figureright figurec

# Argyres-Douglas Theories, IR N-ality and Complete Graphs

Anindya Dey<EMAIL_ADDRESS>
###### Abstract

We show that for a large subclass of Argyres-Douglas-type theories, the Higgs
branch admits multiple hyperkähler quotient realizations as Higgs branches of
three dimensional ${\cal N}=4$ quiver gauge theories, which are related by a
sequence of Seiberg-like IR dualities. We refer to this phenomenon as the
Hyperkähler Quotient N-ality of the four dimensional Higgs branch. The
associated set of 3d theories contains a special subset of maximal unitary
quivers: quiver gauge theories for which the resolution/deformation parameters
of the Higgs branch are manifest in the Lagrangian as Fayet-Iliopoulas
parameters. Starting from the Type IIB description for a given SCFT, we
present an explicit construction to determine the aforementioned set of 3d
quivers, including the subset of maximal unitary quivers. As a byproduct, we
find a simple method for constructing the three dimensional mirror associated
with the SCFT. We demonstrate the construction for the $(A_{k},A_{k})$
theories of Cecotti, Neitzke and Vafa, focusing on the cases $k=3$ and $k=4$.
The associated maximal unitary quiver is unique up to field redefinitions and
turns out to be an Abelian quiver gauge theory. The three dimensional mirror
obtained in this fashion reproduces the well-known complete graph. In the
appendices to the main paper, we study the quotient N-ality in the closely
related family of $D^{b}_{p}(SU(N))$ SCFTs, for which both the maximal unitary
quiver as well as the 3d mirror turn out to be non-Abelian gauge theories
generically.

##  I Introduction

Four dimensional ${\cal N}=2$ Superconformal Field Theories (SCFTs) for which
the Coulomb branch (CB) operators have fractional scaling dimensions are
collectively referred to as Argyres-Douglas (AD) Argyres  _et al._ (1996);
Argyres and Douglas (1995) theories. While the earliest examples of such SCFTs
were obtained as low energy effective field theories at special points on the
moduli space of asymptotically free quiver gauge theories, the program of
Geometric Engineering – pioneered by the papers Klemm  _et al._ (1996); Katz
_et al._ (1997) – gives an efficient algorithm to construct a large class of
these SCFTs. A particularly interesting subclass of such theories can be
obtained by compactifying Type IIB String Theory on an isolated hypersurface
singularity $X$ in four complex dimensions Shapere and Vafa (1999) :

$X=\\{\bm{x}=(x_{1},x_{2},x_{3},x_{4})\in\mathbb{C}^{4}|F(\bm{x})=0\\},$ (1)

where $F(\bm{x})$ is a quasi-homogenous polynomial with an isolated
singularity at the origin, and has the following $\mathbb{C}^{\star}$-action
with weights $q_{i}>0\,\forall i$:

$F(\lambda^{q_{i}}\,x_{i})=\lambda\,F(\bm{x})\quad{\rm
with}\quad\sum_{i}q_{i}>1.$ (2)

The Seiberg-Witten (SW) curve of the 4d SCFT arises from the deformed
singularity:

$\widehat{F}(\bm{x})=F(\bm{x})+\sum_{i,j,k,l}\,u_{ijkl}\,x^{i}_{1}\,x^{j}_{2}\,x^{k}_{3}\,x^{l}_{4}=0,$
(3)

where the coefficients $\\{u_{ijkl}\\}$ include the Coulomb branch operators,
marginal and relevant couplings, and mass parameters associated with the
flavor symmetry of the SCFT. The scaling dimensions of the Coulomb branch
operators and the various couplings can also be read off from the above
equation.

In the Geometric Engineering picture, the CB physics of the 4d SCFT is encoded
in the SW curve which in turn is completely determined by the classical
geometry of the deformed singularity. On the other hand, the 4d Higgs branch
(HB) turns out to be a more subtle object to study, given that the metric
receives non-trivial quantum corrections from the D-brane instantons. There
exists a large literature on the various approaches for studying the 4d HB of
an AD theory – we refer the reader to Argyres  _et al._ (2012); Gaiotto  _et
al._ (2010); Cecotti  _et al._ (2010); Nanopoulos and Xie (2011); Closset  _et
al._ (2021); Giacomelli  _et al._ (2021) for a partial list. One such
approach, possibly the most direct, is to consider an appropriate circle
reduction of the 4d SCFT and flow to a 3d IR SCFT. Since the 4d HB is not
affected by this reduction, one may identify it with the 3d HB of this IR
SCFT. In addition, there may exist a 3d ${\cal N}=4$ quiver gauge theory which
flows to the aforementioned SCFT in the IR, with or without an exchange of the
CB and the HB. In the former case, the 4d HB can be realized as the 3d CB of
the quiver gauge theory – a hyperkähler manifold but not necessarily a
hyperkähler quotient. For the latter case, we have a realization of the 4d HB
as a hyperkähler quotient in the standard fashion from the quiver
representation.

In this paper, we show that for a large class of AD-type SCFTs, the 4d HB
generically admits $N\geq 2$ realizations as hyperkähler quotients, where each
quotient realization is associated with the HB of a 3d ${\cal N}=4$ quiver
gauge theory. We refer to this phenomenon as the hyperkähler quotient N-ality
for a given 4d HB. We also show that this quotient N-ality directly follows
from three dimensional IR N-ality Dey (2023a, 2022) : a set of 3d ${\cal N}=4$
quiver gauge theories flowing to the same IR SCFT without an exchange of the
CBs and the HBs. As discussed in Dey (2023a, 2022), the theories in the N-al
set can be related by a sequence of quiver mutations. The principal objective
of this paper is to present an explicit construction of this N-al set of
quivers for a given 4d SCFT, starting from its Seiberg-Witten geometry.

The number of Fayet-Iliopoulas (FI) parameters that can be turned on for a 3d
quiver is given by the UV-manifest rank of the CB global symmetry. A generic
quiver in the N-al set, however, has emergent CB symmetry : the rank of the IR
symmetry is greater than the rank of the UV-manifest symmetry, which implies
that the IR SCFT has mass deformations that are not visible in the Lagrangian.
Since the FI parameters resolve/deform the HB geometry, a generic quiver
therefore only makes some of the resolution/deformation parameters manifest.
An important result of our paper is to show that there always exists a subset
of quivers in the N-al set for which all the resolution/deformation parameters
of the HB become manifest as FI parameters. We will refer to these quivers as
maximal unitary quivers, and construct them explicitly for certain families of
4d SCFTs. Finally, we will show that our construction leads to a simple method
for computing the 3d mirror of the SCFT.

We will first demonstrate our construction using the $(A_{k},A_{k})$ SCFTs
Cecotti  _et al._ (2010) which arise from the following hypersurface
singularity:

$X=\\{\bm{x}\in\mathbb{C}^{4}|x^{2}_{1}+x^{2}_{2}+x^{k+1}_{3}+x^{k+1}_{4}=0\\}.$
(4)

We will also study the $D^{b}_{p}(SU(N))$ theories Cecotti and Del Zotto
(2013); Cecotti  _et al._ (2013) which are engineered by compactifying Type
IIB String Theory on a hypersurface singularity in
$\mathbb{C}^{3}\times\mathbb{C}^{\star}$ :

$X=\begin{cases}\\{(\bm{x},z)|x^{2}_{1}+x^{2}_{2}+x^{N}_{3}+z^{p}=0\\}\quad
b=N,\\\ \\{(\bm{x},,z)|x^{2}_{1}+x^{2}_{2}+x^{N}_{3}+x_{3}z^{p}=0\\}\quad
b=N-1.\end{cases}$ (5)

where $(\bm{x},z)\in\mathbb{C}^{3}\times\mathbb{C}^{\star}$ denote the local
coordinates. Our construction, however, applies to a much larger class of
SCFTs, as we will specify momentarily. A larger set of examples as well as
additional details of the construction will be discussed in a longer version
of this paper Dey .

The rest of the paper is organized as follows. In Section II, we will give a
brief review of the circle reduction of a 4d SCFT, and discuss certain
S-duality frames of the $(A_{k},A_{k})$ SCFTs which will be relevant for our
construction. In Section III, we will discuss the basic recipe for
constructing the N-al set of 3d quivers and in particular, the maximal unitary
quivers. We will determine the N-al set for the $(A_{k},A_{k})$ SCFTs in
Section IV, explicitly working out the $k=3$ and $k=4$ cases. The general
procedure of constructing the 3d mirror from a maximal unitary quiver will be
discussed in Section V, which will be used to determine the 3d mirrors of the
$(A_{k},A_{k})$ SCFTs in Section VI. The details of quotient N-ality for the
$D^{b}_{p}(SU(N))$ theories can be found in Appendix B – Appendix D. We
summarize our results and comment on future directions of work in Section VII.

$D_{2}(SU(3))$$D_{2}(SU(5))$$D_{2}(SU(k))$$D_{2}(SU(k))$$D_{2}(SU(5))$$D_{2}(SU(3))$
2 3 $\frac{k+1}{2}$ 1 3 2
---
$D_{2}(SU(3))$$D_{2}(SU(5))$$D_{2}(SU(k-1))$$D_{2}(SU(k+1))$$D_{2}(SU(k-1))$$D_{2}(SU(5))$$D_{2}(SU(3))$
2 3 $\frac{k}{2}$ $\frac{k}{2}$ 3 2
Figure 1: An S-duality frame for the $(A_{k},A_{k})$ SCFT for $k$ odd (top)
and $k$ even (bottom) with $k>2$.

##  II Circle Reduction and S-duality frames

Let us begin by discussing the circle reduction of an isolated 4d SCFT i.e. an
SCFT for which the associated conformal manifold is zero-dimensional,
following the treatment in Closset  _et al._ (2021). The first step is to
realize the SCFT as the low energy effective theory at the origin of the CB of
a 4d Lagrangian theory ${\cal K}$ with appropriately scaled mass deformations.
In the next step, one compactifies the theory on $\mathbb{R}^{1,2}\times
S^{1}_{R}$ and considers the limit $R\to 0$ for the energy scales
$E\ll\Lambda$, holding the 4d RG scale
$\Lambda=\exp{(-\frac{8\pi^{2}}{g^{2}_{4}})}$ fixed. This limit corresponds to
the strong-coupling limit of the 3d gauge coupling :
$g^{2}_{3}=\frac{g^{2}_{4}}{2\pi R}\to\infty$. By studying the low energy
effective theory in the neighborhood of the origin of the CB, one can often
realize the resultant 3d SCFT as the IR SCFT of a 3d Lagrangian field theory
${\cal T}$ with no mass deformations at its CB origin. Since the 4d HB remains
unaffected by the circle reduction, it can be realized as the HB of the theory
${\cal T}$ as a hyperkähler quotient.

For a generic non-isolated 4d SCFT, one can implement a similar program if the
SCFT admits an S-duality frame where the SCFT can be described as a network of
isolated SCFTs connected by conformally gauged vector multiplets in the
following fashion :

${\cal T}^{4d}_{1}$${\cal T}^{4d}_{2}$${\cal T}^{4d}_{3}$ $G_{1}$ $G_{2}$

A larger gray node labelled ${\cal T}^{4d}_{i}$ denotes an isolated 4d SCFT
while a smaller gray node labelled $G_{i}$ denotes a conformally gauged 4d
vector multiplet with gauge group $G_{i}$. An isolated SCFT connected to a
$G_{i}$ vector multiplet must have a flavor symmetry subgroup $G_{i}$. The
number of vector multiplets matches the number of marginal couplings (or
equivalently, the dimension of the conformal manifold) of the SCFT. The above
description of the SCFT is at times called a partially weakly-coupled
description.

Given such an S-duality frame, one can again compactify the theory on a circle
and reduce to 3d. The conformally gauged vector multiplet descends to a 3d
vector multiplet of the same gauge group, while the 3d Lagrangian theories
corresponding to the isolated SCFTs can be derived as before. Putting the
components together, one therefore has a 3d ${\cal N}=4$ quiver gauge theory
of the following form:

${\cal T}^{3d}_{1}$${\cal T}^{3d}_{2}$${\cal T}^{3d}_{3}$ $G_{1}$ $G_{2}$

A larger gray node labelled ${\cal T}^{3d}_{i}$ denotes a 3d quiver while a
smaller gray node denotes a 3d $G_{i}$ vector multiplet. The 4d HB is then
given by the HB of this 3d ${\cal N}=4$ quiver gauge theory and is therefore
realized as a hyperkähler quotient.

The construction presented in this paper will apply to the subclass of 4d
SCFTs which admits a partially weakly-coupled description as described above,
with the constraint that the gauge group for any conformally gauged vector
multiplet has to be special unitary. In addition, we will restrict ourselves
to the subclass of isolated SCFTs for which the associated 3d quivers ${\cal
T}^{3d}_{i}$ are given by unitary quiver gauge theories with hypermultiplets
in the fundamental and the bifundamental representations. The resultant 3d
quiver has the generic form shown in Figure 2, which precisely belongs to the
class that was studied in the context of IR N-ality in the papers Dey (2023a,
2022). To distinguish it from other 3d quivers that we will discuss later in
the paper, we will refer to this quiver as the U-SU quiver associated with the
4d SCFT.

$N_{1}$ $N_{2}$ $N_{\alpha}$ $N_{\alpha+1}$ $N_{\alpha+2}$ $M_{\alpha+2}$
$M_{1}$ $M_{2}$ $M_{\alpha}$ $M_{\alpha+1}$

Figure 2: A generic quiver with unitary/special unitary gauge nodes and
(bi)fundamental matter. The yellow and the black nodes denote the special
unitary vector multiplets and the unitary vector multiplets respectively, with
the black lines connected to gauge nodes denote fundamental/bifundamental
hypermultiplets.

As an explicit example, we will write down the relevant S-duality frames of
the $(A_{k},A_{k})$ SCFTs, which have been studied in the literature from the
perspective of Hitchin systems Xie and Yau (2017) as well as that of the Type
IIB realization Buican  _et al._ (2015); Giacomelli  _et al._ (2021). To do
this, we will need the $D^{b}_{p}(SU(N))$ theories – in particular, the
subfamily of isolated SCFTs $D_{2}(SU(2n-1))$ for an integer $n>1$. The
relevant S-duality frame for an $(A_{k},A_{k})$ SCFT is shown in Figure 1, for
$k$ odd and $k$ even respectively. Note that there are precisely $k-2$ special
unitary gauge nodes corresponding to the number of marginal couplings of the
SCFT. The circle reduction of the $D_{2}(SU(2n-1))$ is discussed in Appendix B
: the resultant 3d SCFT can be realized as the SCFT at the origin of the CB of
a $U(n-1)$ SQCD with $N_{f}=2n-1$. Combining this result with the S-duality
frame in Figure 1, one can write down the U-SU quivers for the $(A_{k},A_{k})$
SCFTs – given in Figure 3 for $k$ odd and $k$ even respectively. These quivers
are linear quivers with unitary and special unitary gauge nodes and are
therefore a special subclass of the quivers in Figure 2. In the next section,
we will discuss how the hyperkähler quotient N-ality arises from these U-SU
quivers.

1 1 2 2 3 $\frac{k-1}{2}$ $\frac{k+1}{2}$ 1 $\frac{k-1}{2}$ 3 2 2 1 1
---
1 1 2 2 3 $\frac{k}{2}-1$ $\frac{k}{2}$ $\frac{k}{2}$ $\frac{k}{2}$
$\frac{k}{2}-1$ 3 2 2 1 1 1
Figure 3: The U-SU quivers for the $(A_{k},A_{k})$ SCFT for $k$ odd (top) and
$k$ even (bottom) with $k>2$.

##  III The Basic Construction : Duality Sequence and the Maximal Unitary
Quiver

Consider a U-SU quiver of the generic form given in Figure 2. We define a
balance parameter $e_{\alpha}$ for a given $U(N_{\alpha})$ or $SU(N_{\alpha})$
gauge node as $e_{\alpha}=N^{\alpha}_{f/bf}-2N_{\alpha}$, where
$N^{\alpha}_{f/bf}$ denotes the total number of fundamental and bifundamental
hypers associated with the gauge node. We refer to an $U(N_{\alpha})$ node as
balanced if $e_{\alpha}=0$, while an $SU(N_{\alpha})$ node is balanced if
$e_{\alpha}=-1$. Given this convention, we note that all of the $k-2$ special
unitary nodes in the linear U-SU quiver associated with the $(A_{k},A_{k})$
SCFT are balanced, while the unitary nodes are all overbalanced. One can
additionally check that these quiver gauge theories are good theories in the
Gaiotto-Witten sense.

$N_{1}$ $N_{2}$ $N_{\alpha}$ $N_{\alpha+1}$ $M_{\alpha\,\alpha+1}$
$N_{\alpha+2}$ $M_{\alpha+2}$ $F$ $M_{1}$ $M_{2}$ $M_{\alpha}$
$(\widetilde{Q}^{1},\widetilde{Q}^{2})$ $P$ $Q^{1}$ $Q^{2}$ $Q^{\alpha}$

Figure 4: A generic quiver with unitary/special unitary gauge nodes, with
(bi)fundamental and Abelian hypermultiplets. A blue box with label $F$ denotes
$F$ Abelian hypermultiplets in the determinant representation, while blue
lines connecting multiple gauge nodes denote hypermultiplets charged under
those gauge nodes with charges $\\{Q^{i}\\}$.

It was shown in Dey (2023a, 2022) that a U-SU quiver ${\cal T}$ with at least
one balanced SU gauge node has an emergent CB global symmetry:

$\text{rk}(\mathfrak{g}^{\rm IR}_{\rm C}({\cal
T}))>\text{rk}(\mathfrak{g}^{\rm UV}_{\rm C}({\cal T})),$ (6)

where $\mathfrak{g}^{\rm IR/UV}_{\rm C}({\cal T})$ denotes the CB symmetry
algebra for the theory ${\cal T}$ in the IR/UV. The UV symmetry algebra is
simply given by the direct sum of the $\mathfrak{u}(1)$ topological symmetries
i.e. $\mathfrak{g}^{\rm UV}_{\rm C}({\cal
T})=\oplus^{L}_{\alpha=1}\,\mathfrak{u}(1)_{\alpha}$, with $\alpha$ labelling
the unitary gauge nodes in ${\cal T}$. The existence of an emergent CB global
symmetry implies that the 3d IR SCFT has mass deformations which are not
visible in the quiver as FI parameters. This in turn implies that for the
hyperkähler quotient constructed using the U-SU quiver, one can turn on only a
subset of resolution/deformation parameters of the HB geometry. It was also
shown in Dey (2022) that a U-SU quiver with an emergent symmetry admits a
sequence of IR dualities, generated by a set of four quiver mutations –
mutations $I$, $I^{\prime}$, $II$ and $III$ – which obey non-trivial closure
relations. We review these mutations in Appendix A and refer the reader to the
papers Dey (2023a, 2022) for further details.

Each IR duality in this sequence maps the CB (HB) of one theory to the CB (HB)
of the dual theory in the IR. The duality sequence therefore generates a set
of quiver gauge theories which flow to the same SCFT in the IR without an
exchange of the CB and the HB – this is dubbed as IR N-ality and the
aforementioned set of theories is referred to as the N-al set. A generic
theory in the N-al set, however, is not a U-SU quiver of the form given in
Figure 2 but it is a more general quiver gauge theory of the form given in
Figure 4. In addition to unitary/special unitary gauge nodes and
fundamental/bifundamental matter, the theory in Figure 4 has hypermultiplets
that transform in powers of the determinant and/or the anti-determinant
representations of the unitary gauge nodes. We refer to these matter
multiplets collectively as Abelian hypermultiplets. Evidently, the class of
quivers in Figure 2 is a subclass of the quivers in Figure 4.

As reviewed in Appendix A, the quiver mutation $I$ acts on a balanced
$SU(N_{\alpha})$ gauge node, while remaining mutations act on unitary gauge
nodes connected to Abelian hypermultiplet(s). The mutations
$I^{\prime},II,III$ act on such a $U(N_{\alpha})$ gauge node with balance
parameters $e_{\alpha}=1,0,-1$ respectively. Given the quiver mutations, the
recipe for writing down the 3d quivers associated with a 4d SCFT is
straightforward. One first implements the circle reduction discussed in
Section II to determine the U-SU quiver gauge theory ${\cal T}$ of the generic
form given in Figure 2, which will generically have $l\geq 1$ balanced SU
gauge nodes. Given the quiver ${\cal T}$, one implements mutation $I$ at every
balanced SU node giving rise to $l$ distinct quiver gauge theories. Each such
gauge theory will have $l-1$ balanced SU nodes as well as unitary nodes which
are connected to an Abelian hypermultiplet and have different balance
parameters $e_{\alpha}\geq-1$. In the next step, one implements all admissible
quiver mutations on this gauge gauge theory to generate another set of quiver
gauge theories. One continues this procedure until no new quiver gauge theory
can be generated using the above mutations.

The quiver mutations $I$ and $III$ increase the number of $\mathfrak{u}(1)$
topological symmetries (equivalently the number of FI parameters) by 1. On the
other hand, mutation $I^{\prime}$ decreases this number by 1, and mutation
${II}$ keeps it invariant. Therefore, the 3d Lagrangians generated by the
quiver mutations will generically differ among themselves in UV-manifest rank
of the CB global symmetry. This implies that the associated hyperkähler
quotient realizations will differ in the number of resolution/deformation
parameters that can be turned on. The U-SU quiver ${\cal T}$ has the minimum
number of FI parameters, since there does not exist a quiver mutation that can
decrease the number of topological symmetries further on this quiver. At the
other extreme, there exists a subset of quivers for which the number of FI
parameters precisely matches the number of resolution/deformation parameters
for the 4d HB. We will call a generic quiver in this subset a maximal unitary
quiver and denote it as ${\cal T}_{\rm maximal}$ for a given U-SU quiver
${\cal T}$. In the duality sequence discussed above, a quiver can be
identified as maximal unitary if none of its gauge nodes admit either a
mutation $I$ or a mutation $III$. A distinct pair of maximal unitary quivers
are related by a sequence of mutation $II$.

1 1 2 1 1 1 $({\cal T})$ |  |  ${\cal O}_{I}$ ${\cal O}_{I^{\prime}}$ |  |  $(1,-1,1)$ 1 1 1 1 1 1 $({\cal T}_{\rm maximal})$
---|---|---|---|---
Figure 5: The duality sequence and the 3d quivers associated with the
$(A_{3},A_{3})$ SCFT. For every quiver, the nodes which admit a mutation is
marked by a cross.

If all the SU nodes in ${\cal T}$ are balanced, the theory ${\cal T}_{\rm
maximal}$ consists only of unitary gauge nodes with fundamental/bifundamental
hypers as well as Abelian hypermultiplets. In addition, it has the property
that every non-Abelian unitary node is either balanced or overbalanced. For
the $(A_{k},A_{k})$ SCFTs, the theory ${\cal T}_{\rm maximal}$ turns out to be
a unique Abelian quiver gauge theory.

The fact that the number of FI parameters in ${\cal T}_{\rm maximal}$ matches
the number of resolution/deformation parameters of the 4d HB can be verified
from the 3d mirror. Under mirror symmetry, the HB of ${\cal T}_{\rm maximal}$
is mapped to the CB of the 3d mirror in the deep IR, and in particular the FI
parameters of the former are mapped to the mass parameters of the latter.
Since turning on generic mass parameters completely resolves the CB of the 3d
mirror as an algebraic variety, it follows from the above argument that the HB
of ${\cal T}_{\rm maximal}$ (and equivalently the 4d HB) is also completely
resolved/deformed by turning on generic FI parameters. Therefore, one simply
needs to check that the number of FI parameters in ${\cal T}_{\rm maximal}$
matches the number of independent mass parameters in the 3d mirror. We will
perform this check explicitly for the $(A_{k},A_{k})$ SCFTs in the later
sections.

##  IV Hyperkähler Quotient N-ality : $(A_{k},A_{k})$ SCFTs

Let us now demonstrate how the general construction presented in Section III
works for the $(A_{k},A_{k})$ SCFTs. We begin with the simplest SCFT with a
non-trivial conformal manifold – the $(A_{3},A_{3})$ theory – which has a
conformal manifold of dimension one. From the top figure in Figure 1, the
S-duality frame can be seen to have the following form:

$D_{2}(SU(3))$$D_{2}(SU(3))$ 2 1

The U-SU quiver for the $(A_{3},A_{3})$ SCFT can be read off from the
S-duality frame as described in Section II, and has the following form:

$[(A_{3},A_{3})]_{\rm 3d}$: |  |  1 1 2 1 1 1
---|---|---

The central $SU(2)$ node is balanced and one can therefore implement mutation
$I$ at this node as shown in Figure 5. The resultant quiver is an Abelian
gauge theory, and none of the gauge nodes admit either mutation $I$ or
mutation $III$. Therefore, this quiver can be identified as the maximal
unitary quiver. The central $U(1)$ node in ${\cal T}_{\rm maximal}$ has
balance parameter $e=1$ and therefore admits a mutation $I^{\prime}$ which
takes ${\cal T}_{\rm maximal}$ back to the quiver ${\cal T}$. In this case,
the N-al set consists of two distinct Lagrangians. Therefore, the hyperkähler
quotient N-ality associated with the $(A_{3},A_{3})$ SCFT is simply a duality.
Note that $\text{rk}(\mathfrak{g}^{\rm UV}_{\rm C}({\cal T}_{\rm maximal}))=3$
corresponding to the three unitary gauge nodes, while
$\text{rk}(\mathfrak{g}^{\rm UV}_{\rm C}({\cal T}))=2$. One can therefore turn
on 3 FI parameters in ${\cal T}_{\rm maximal}$, while for ${\cal T}$ one can
turn on only 2.

1 1 2 2 2 1 1 1 $({\cal T})$ |  ${\cal O}_{I}$ ${\cal O}_{I^{\prime}}$ |  $(1,-1,2)$ 1 1 1 2 2 1 1 1 $({\cal T}_{1})$
---|---|---
|  |  ${\cal O}_{I}$ ${\cal O}_{I^{\prime}}$
$(1,1,-1,1)$ $(1,1,-1,1)$ 1 1 1 1 1 1 1 1 1 1 $({\cal T}_{\rm maximal})$ |  ${\cal O}_{III}$ ${\cal O}_{I^{\prime}}$ |  $(1,-1,2)$ $(2,-1,1)$ 1 1 1 2 1 1 1 1 $({\cal T}_{2})$
Figure 6: The duality sequence and the 3d quivers for the $(A_{4},A_{4})$
SCFT. For every quiver, the nodes which admit a mutation is marked by a cross.

Let us consider the next non-trivial example – the $(A_{4},A_{4})$ SCFT, for
which the conformal manifold has dimension 2. From the bottom figure in Figure
1, the S-duality frame can be seen to have the following form:

$D_{2}(SU(3))$$D_{2}(SU(5))$$D_{2}(SU(3))$ 2 2

The U-SU quiver for the $(A_{4},A_{4})$ SCFT can be read off from the
S-duality frame described in Section II, and has the following form:

$[(A_{4},A_{4})]_{\rm 3d}$: |  |  1 1 2 2 2 1 1 1
---|---|---

The theory has two balanced $SU(2)$ gauge nodes corresponding to the two
marginal couplings in the SCFT, while the unitary nodes are overbalanced. Also
note that the quiver is symmetric about the central $U(2)$ gauge node.
Following the prescription in Section III, the 3d quivers constituting the
N-al set can be found starting from the above quiver by the sequence of quiver
mutations shown in Figure 6:

  1. 1.

We begin by implementing mutation $I$ at the $SU(2)$ node on the left which
introduces an Abelian hypermultiplet. The resultant quiver ${\cal T}_{1}$ has
a single balanced $SU(2)$ node. Since the quiver ${\cal T}$ is symmetric about
the central $U(2)$ node, implementing mutation $I$ on the right $SU(2)$ node
will produce the same quiver up to a discrete identification.

  2. 2.

The quiver ${\cal T}_{1}$ admits mutation $I$ at the remaining balanced
$SU(2)$ node which introduces a second Abelian hypermultiplet and gives the
quiver ${\cal T}_{2}$. The second $U(1)$ node from the left has balance
parameter $e=1$ and therefore admits a mutation $I^{\prime}$. This mutation
takes the quiver ${\cal T}_{1}$ back to the quiver ${\cal T}$. It might appear
that the central $U(2)$ node of ${\cal T}_{1}$ admits a mutation $II$ but one
can check that this operation amounts a change of variables of the Abelian
vector multiplets and therefore does not produce a distinct Lagrangian.

  3. 3.

In the next step, we observe that the quiver ${\cal T}_{2}$ has only unitary
gauge nodes – all of which are $U(1)$ nodes except for the central $U(2)$
gauge node which has balance parameter $-1$. The $U(2)$ node also connected to
the two Abelian hypermultiplets and therefore admits a mutation $III$.
Implementing this mutation gives an Abelian quiver gauge theory. This quiver
does not admit mutation $I$ or $III$, and it can therefore be identified as
the maximal unitary quiver. The second and the fourth gauge nodes from the
left (both $U(1)$ nodes) have balance parameters $e=1$ and admit mutation
$I^{\prime}$. Implementing it at any of these nodes takes the quiver ${\cal
T}_{2}$ back to the quiver ${\cal T}_{1}$.

  4. 4.

In the quiver ${\cal T}_{\rm maximal}$, one of the $U(1)$ gauge nodes (marked
with a cross) connected to the Abelian hypers has balance parameter $e=1$ and
therefore admits a mutation $I^{\prime}$. This mutation takes ${\cal T}_{\rm
maximal}$ back to the quiver ${\cal T}_{2}$.

Therefore, the N-al set associated with the $(A_{4},A_{4})$ SCFT consists of
four distinct quiver gauge theories. The hyperkähler quotient N-ality in this
case is a quadrality. Note that the UV-manifest rank of the CB global symmetry
(or equivalently the number of FI parameters) in the quiver ${\cal T}$ is 3,
and it increases by 1 with every mutation $I$ and $III$ reaching the maximum
of 6 for ${\cal T}_{\rm maximal}$.

One can now see how the duality sequence works out for the case of a generic
$k$. The U-SU quiver has $k-2$ balanced SU nodes and $k-1$ overbalanced
unitary nodes. Mutation $I$ at the balanced nodes will bring a total of $k-2$
Abelian hypermultiplets, while applying Mutation $III$ sequentially leads to a
maximal unitary quiver which is an Abelian gauge theory. The general structure
of the quiver ${\cal T}_{\rm maximal}$ is given as follows. One starts from
the linear quiver with $2k-3$ $U(1)$ nodes:

1 1 1 1 1 1 1 1

(7)

with the central $U(1)$ gauge node ($(k-1)$-th from the left or the right)
having a $U(1)$ flavor node. The quiver ${\cal T}_{\rm maximal}$ is then given
by decorating the linear quiver with a total of $(k-2)$ Abelian
hypermultiplets, which may in turn be attached to Abelian quiver tails. The
precise charges of the Abelian hypers and the quiver tails may be worked out
using the duality sequence for any given $k$.

##  V From the Maximal Unitary Quiver to the 3d Mirror

In this section, we write down the general recipe for constructing the 3d
mirror of an AD-type SCFT, focusing on the subclass of theories for which the
circle reduction of the SCFT gives a U-SU quiver ${\cal T}$, which is a good
theory in the Gaiotto-Witten sense. In addition, ${\cal T}$ obeys the
constraint that all the special unitary nodes are balanced. We will follow the
treatment of Dey (2023b) where the problem of finding the 3d mirror was
studied for a generic good 3d U-SU quiver gauge theory with at least one
balanced special unitary node.

The starting point of the construction is the maximal unitary quiver ${\cal
T}_{\rm maximal}$ (or one of the maximal unitary quivers) which can be derived
from ${\cal T}$ using a sequence of the quiver mutations $I,III$ and $II$, as
discussed above. Since mutation $I$ introduces an Abelian hyper and mutation
$III$ introduces an Abelian hyper attached to a quiver tail, the quiver ${\cal
T}_{\rm maximal}$ always has the following general structure – it is given by
a standard quiver gauge theory ${\cal T}_{\rm good}$ consisting of unitary
gauge nodes and matter in the fundamental/bifundamental representation,
decorated with Abelian hypermultiplets attached to Abelian quiver tails. For
the $(A_{k},A_{k})$ SCFT, the quiver ${\cal T}_{\rm good}$, given by (7), is
particularly simple – a linear quiver with $U(1)$ gauge nodes which is a good
theory. From the general rules of quiver mutations, one can check that the
quiver ${\cal T}_{\rm good}$ is at most an ugly theory, if ${\cal T}$ is a
good theory to begin with.

Given this general structure of ${\cal T}_{\rm maximal}$, the 3d mirror can be
found in the following fashion. In the first step, the Abelian hypers and the
Abelian quiver tails introduced by the quiver mutations are stripped off from
${\cal T}_{\rm maximal}$ to obtain the quiver ${\cal T}_{\rm good}$. Next, we
find the 3d mirror of ${\cal T}_{\rm good}$, which will be denoted as
$\widetilde{{\cal T}}_{\rm good}$. If ${\cal T}_{\rm good}$ is a linear quiver
with unitary gauge nodes, $\widetilde{{\cal T}}_{\rm good}$ will also belong
to the same class. It will also be a good theory, plus a certain number of
free hypermultiplets if ${\cal T}_{\rm good}$ has ugly nodes.

The theory ${\cal T}_{\rm maximal}$ can be built up from of the theory ${\cal
T}_{\rm good}$ by a sequence of simple QFT operations, each of which can be
thought of as a mild generalization of the $S$-operation Witten (2003).
Generically, if the theory ${\cal T}$ has $l$ balanced special unitary nodes,
the quiver ${\cal T}_{\rm maximal}$ consists of $l$ Abelian hypermultiplets.
Starting from ${\cal T}_{\rm good}$, we first add to the theory ${\cal T}_{\rm
good}$ a decoupled Abelian quiver gauge theory ${\cal T}^{(1)}_{\rm
decoupled}$ of the following form:

${\cal T}^{(1)}_{\rm decoupled}:$ 1 1 1 1 1

where the number of $U(1)$ gauge nodes is $(s+1)$. Evidently, this quiver is
simply another way of denoting a collection of $(s+1)$ twisted free
hypermultiplets. Then, we gauge an $\mathfrak{u}(1)$ subalgebra of the
topological symmetry algebra of the combined theory corresponding to the
following generator :

$J=J^{(1)}_{(s+1)}+\sum_{a}\,\frac{Q_{a}}{N_{a}}\,J_{a},$ (8)

where $J^{(1)}_{(i)}$ is the generator for the topological symmetry associated
with the $i$-th $U(1)$ gauge node of the ${\cal T}^{(1)}_{\rm decoupled}$ ($i$
increasing from left to right), while $J_{a}$ is the topological symmetry
generator associated with the $U(N_{a})$ gauge node in ${\cal T}_{\rm good}$.
The sum on the RHS extends over all nodes of the quiver ${\cal T}_{\rm good}$.
The result of this operation is to attach an Abelian hyper to a collection of
gauge nodes $\\{U(N_{a})\\}$ in ${\cal T}_{\rm good}$ with respective charges
$\\{Q_{a}\\}$, where the Abelian hyper is connected to a quiver tail of $s$
$U(1)$ gauge nodes. Also, the charge of the Abelian hyper under the $U(1)$
node of the quiver tail it is connected to is 1. These type of QFT operations
(and certain generalized versions) were studied in detail in Dey (2021) and
were used to construct new pairs of 3d mirrors from a given mirror pair. We
refer the reader to Section 4.1 of Dey (2021) for further details.

Let us call the quiver obtained by the above QFT operation ${\cal
T}^{(1)}_{\rm good}$, which is a unitary gauge theory with a single Abelian
hypermultiplet. In the next step, one performs a similar operation now
starting with ${\cal T}^{(1)}_{\rm good}$, adding a decoupled theory ${\cal
T}^{(2)}_{\rm decoupled}$ of the same form as before and then gauging a
$\mathfrak{u}(1)$ subalgebra of the topological symmetry of the combined
theory. This results in the quiver ${\cal T}^{(2)}_{\rm good}$ which is a
unitary gauge theory with two Abelian hypers. Repeating the operation $l$
times, one obtains ${\cal T}^{(l)}_{\rm good}={\cal T}_{\rm maximal}$, which
is quiver with $l$ Abelian hypermultiplets attached to Abelian quiver tails.

On the mirror side, one implements the dual sequence of QFT operations as
follows. Starting from $\widetilde{{\cal T}}_{\rm good}$, the first QFT
operation involves adding a decoupled sector $\widetilde{{\cal T}}^{(1)}_{\rm
decoupled}$ of free hypermultiplets – the mirror dual of ${\cal T}^{(1)}_{\rm
decoupled}$ – which we represent as :

$\widetilde{{\cal T}}^{(1)}_{\rm decoupled}:$ 1 $s+1$

.

Under mirror symmetry, the $\mathfrak{u}(1)$ subalgebra in (8) maps to a
$\mathfrak{u}(1)$ subalgebra of the HB global symmetry algebra for the
combined theory $\widetilde{{\cal T}}_{\rm good}\oplus\widetilde{{\cal
T}}^{(1)}_{\rm decoupled}$. The QFT operation then amounts to identifying a
set of $U(1)$ flavor nodes in $\widetilde{{\cal T}}_{\rm good}$ with the
$U(1)$ flavor node in $\widetilde{{\cal T}}^{(1)}_{\rm decoupled}$, and then
gauging the identified flavor node. This operation is an example of an Abelian
identification-flavoring-gauging operation introduced in Dey (2021). The
resultant quiver is the 3d mirror of ${\cal T}^{(1)}_{\rm good}$ and we denote
it by $\widetilde{{\cal T}}^{(1)}_{\rm good}$. In the next step, we perform a
similar operation starting with $\widetilde{{\cal T}}^{(1)}_{\rm good}$. After
performing $l$ such operations, one ends up with the theory $\widetilde{{\cal
T}}^{(l)}_{\rm good}$, which may be identified with $\widetilde{{\cal T}}_{\rm
maximal}$ – the 3d mirror of ${\cal T}_{\rm maximal}$. Given IR N-ality, the
quiver $\widetilde{{\cal T}}_{\rm maximal}$ is the 3d mirror of all the
theories in the N-al set, including the quiver ${\cal T}$. We will explicitly
show how this construction can be used to build the 3d mirrors of the
$(A_{k},A_{k})$ SCFTs.

##  VI 3d mirrors for $(A_{k},A_{k})$ SCFT : Complete graphs

In this section, we will apply the prescription described in Section V now to
construct the 3d mirror of an $(A_{k},A_{k})$ SCFT. Let us start with the case
of $k=3$. The 3d mirror pair $({\cal T}_{\rm good},\widetilde{{\cal T}}_{\rm
good})$ in this case is given as follows:

1 1 1 1 1 1 $({\cal T}_{\rm good})$ |  |  1 1 2 2 $(\widetilde{{\cal T}}_{\rm good})$
---|---|---

Next, we add a theory ${\cal T}_{\rm decoupled}$ – a $U(1)$ theory with a
single hypermultiplet – to the theory ${{\cal T}}_{\rm good}$, while on the
dual side we add the 3d mirror of ${\cal T}_{\rm decoupled}$ – a free
hypermultiplet to the theory $\widetilde{{\cal T}}_{\rm good}$. The QFT
operation on the former side involves gauging a $\mathfrak{u}(1)$ subalgebra
of topological symmetry of the combined theory, where the subalgebra
corresponds to the generator:

$J=J_{0}+J_{1}-J_{2}+J_{3},$ (9)

where $J_{0}$ is the generator of the topological symmetry for the $U(1)$
gauge node in ${\cal T}_{\rm decoupled}$, while $J_{i}$ is the generator of
the topological symmetry for the $i$-th $U(1)$ gauge node in ${{\cal T}}_{\rm
good}$ (with $i$ increasing from left to right). This gives the quiver ${\cal
T}_{\rm maximal}$ :

1 1 1 1 1 1 1 1 $({\cal T}_{\rm good}):$ $({\cal T}_{\rm decoupled}):$
---
$S$
$(1,-1,1)$ 1 1 1 1 1 1 $({\cal T}_{\rm maximal}):$

The red dashed lines mark the gauge nodes whose topological symmetry
generators appear on the RHS of (9). As mentioned in Section V, the operation
on the mirror side therefore involves an identification of the $U(1)$ flavor
nodes as shown below followed by a gauging operation of the identified node:

1 1 1 1 1 1 1 1 $(\widetilde{{\cal T}}_{\rm good}):$ $(\widetilde{{\cal
T}}_{\rm decoupled}):$
---
$S$
1 1 1 1 1 1 $(\widetilde{{\cal T}}_{\rm maximal}):$

The resultant quiver $\widetilde{{\cal T}}_{\rm maximal}$ is equivalent to the
complete graph of with four vertices and edge multiplicity 1, after decoupling
a $U(1)$ vector multiplet associated with one of the vertices:

1 1 1 1 1 1 |  |  1 1 1 1
---|---|---

Given the 3d mirror, one can check that the rank of its HB global symmetry
algebra is $\text{rk}(\mathfrak{g}_{\rm H}(\widetilde{{\cal T}}_{\rm
maximal}))=3$. Therefore, the rank of the HB global symmetry algebra of the
mirror (or equivalently the number of independent mass parameters) precisely
matches the rank of the CB global symmetry algebra of ${\cal T}_{\rm maximal}$
(or equivalently the number of FI parameters). We therefore have an explicit
check that the maximal unitary quiver makes all the resolution/deformation
parameters of the 4d HB manifest in this case.

1 1 1 1 1 1 1 1 1 1 1 $({\cal T}_{\rm good}\oplus{\cal T}^{(1)}_{\rm decoupled}\oplus{\cal T}^{(2)}_{\rm decoupled})$ |  $S_{1}$ |  $(1,1,-1,1)$ 1 1 1 1 1 1 1 1 1 1 $({\cal T}^{(1)}_{\rm good}\oplus{\cal T}^{(2)}_{\rm decoupled})$ |  $S_{2}$ |  $(1,1,-1,1)$ $(1,1,-1,1)$ 1 1 1 1 1 1 1 1 1 1 $({\cal T}_{\rm maximal})$
---|---|---|---|---
1 1 1 2 1 2 1 1 1 2 $(\widetilde{{\cal T}}_{\rm good}\oplus\widetilde{{\cal T}}^{(1)}_{\rm decoupled}\oplus\widetilde{{\cal T}}^{(2)}_{\rm decoupled})$ |  $S_{1}$ |  1 1 1 1 1 1 1 1 1 1 1 $(\widetilde{{\cal T}}^{(1)}_{\rm good}\oplus\widetilde{{\cal T}}^{(2)}_{\rm decoupled})$ |  $S_{2}$ |  1 1 1 1 1 1 1 1 $(\widetilde{{\cal T}}_{\rm maximal})$
Figure 7: Construction of the 3d mirror of the $(A_{4},A_{4})$ SCFT.

Next, consider the case of $(A_{4},A_{4})$ SCFT. In this case, the theory
${\cal T}_{\rm good}$ and its 3d mirror $\widetilde{{\cal T}}_{\rm good}$ are
given as follows:

1 1 1 1 1 1 1 1 $({\cal T}_{\rm good})$ |  |  1 1 3 3 $(\widetilde{{\cal T}}_{\rm good})$
---|---|---

Starting from the mirror pair $({{\cal T}}_{\rm good},\widetilde{{\cal
T}}_{\rm good})$, the 3d mirror of ${\cal T}_{\rm maximal}$ can be generated
by implementing a sequence of two QFT operations which are shown in Figure 7.
To begin with, we add to the theory ${{\cal T}}_{\rm good}$ a decoupled theory
${\cal T}^{(1)}_{\rm decoupled}$ where ${\cal T}^{(1)}_{\rm decoupled}$ is an
Abelian quiver tail of the following form:

${\cal T}^{(1)}_{\rm decoupled}:$ 1 1 1

.

On the dual side, we add to the theory $\widetilde{{\cal T}}_{\rm good}$ the
3d mirror of ${\cal T}^{(1)}_{\rm decoupled}$ – a collection of 2 free
hypermultiplets, which we represent as:

$\widetilde{{\cal T}}^{(1)}_{\rm decoupled}:$ 1 2

.

The operation $S_{1}$ then acts on the combined theory ${\cal T}_{\rm
good}\oplus{\cal T}^{(1)}_{\rm decoupled}$ by gauging a $\mathfrak{u}(1)$
subalgebra of the topological symmetry corresponding to the generator:

$J={J}^{(1)}_{2}+J_{1}-J_{3}+J_{4},$ (10)

where ${J}^{(1)}_{k}$ is the generator of the topological symmetry for the
$U(1)_{k}$ gauge node ($k=1,2$ increasing from left to right) in ${\cal
T}^{(1)}_{\rm decoupled}$, while $J_{i}$ is the generator of the topological
symmetry for the $i$-th $U(1)$ gauge node ($i=1,\ldots,5$ increasing from left
to right) in ${{\cal T}}_{\rm good}$. This operation gives the quiver ${{\cal
T}}^{(1)}_{\rm good}$, as shown in the top row of Figure 7. The operation
$S_{1}$ on the dual side is shown in the bottom row and it leads to the quiver
$\widetilde{{\cal T}}^{(1)}_{\rm good}$.

In the next step, we introduce the second decoupled theory ${{\cal
T}}^{(2)}_{\rm decoupled}$ which is a $U(1)$ theory with a single
hypermultiplet. On the mirror side, we add a free hypermultiplet which is the
3d mirror of ${{\cal T}}^{(2)}_{\rm decoupled}$. The second operation $S_{2}$
then acts on the theory ${{\cal T}}^{(1)}_{\rm good}\oplus{\cal T}^{(2)}_{\rm
decoupled}$ by gauging a $\mathfrak{u}(1)$ subalgebra of the topological
symmetry corresponding to the generator:

$J^{\prime}={J}^{(2)}+{J}^{(1)}_{1}+J_{2}-J_{3}+J_{4},$ (11)

where ${J}^{(2)}$ is the generator of the topological symmetry for the $U(1)$
gauge node in ${\cal T}^{(2)}_{\rm decoupled}$. This operation gives the
theory ${\cal T}_{\rm maximal}$, as shown in the top row of Figure 7. On the
mirror side, the operation gives the quiver $\widetilde{{\cal T}}_{\rm
maximal}$ \- the 3d mirror of ${\cal T}_{\rm maximal}$. The quiver
$\widetilde{{\cal T}}_{\rm maximal}$ can be readily identified as the complete
graph with five vertices and edge multiplicity 1, after decoupling a $U(1)$
vector multiplet associated with one of the vertices:

1 1 1 1 1 1 1 1 |  |  1 1 1 1 1
---|---|---

Given the 3d mirror, one can check that the rank of its HB global symmetry
algebra is $\text{rk}(\mathfrak{g}_{\rm H}(\widetilde{{\cal T}}_{\rm
maximal}))=6$. Therefore, the rank of the HB global symmetry algebra of the
mirror (or equivalently the number of independent mass parameters) precisely
matches the rank of the CB global symmetry algebra of ${\cal T}_{\rm maximal}$
(or equivalently the number of FI parameters). This confirms that the maximal
unitary quiver makes all the resolution/deformation parameters of the 4d HB
manifest.

##  VII Conclusions and Future Directions

We have shown that the 4d HB for a large class of Argyres-Douglas theories
generically admits multiple hyperkähler quotient realizations as Higgs
branches of 3d ${\cal N}=4$ quiver gauge theories. The 3d quivers obey IR
N-ality : they flow to the same 3d SCFT in the deep IR without any exchange of
the CB and the HB, and may be related to each other by a sequence of four
distinct types of quiver mutations. The 3d quivers differ amongst each other
in the UV-manifest rank of the CB global symmetry, but there exists a special
subset of quivers – the maximal unitary quivers – for which the ranks of the
UV-manifest symmetry and the emergent IR symmetry match. For this subset of
quivers, all the resolution/deformation parameters of the HB geometry are
classically manifest as FI parameters. Given the maximal unitary quiver(s),
one can deploy a sequence of Abelian QFT operations to engineer the 3d mirror
of the SCFT.

Starting from the Type IIB description of the 4d SCFT, we have presented a
systematic construction to determine the associated N-al set of 3d quivers
including the maximal unitary ones, leading to the 3d mirror. We should
emphasize that our construction is limited to the subclass of of 4d SCFTs
which after circle reduction, in a way described in Section II, gives a 3d
U-SU quiver (quiver with unitary/special unitary gauge nodes and matter in
fundamental/bifundamental representation) with at least a single balanced
special unitary node. This is the class of theories for which our tools of IR
N-ality and quiver mutations, discussed in Section III, can be applied to
uncover the quotient N-ality. A generic quiver in this set of 3d N-al theories
is a unitary/special unitary quiver gauge theory with Abelian hypermultiplets
– hypermultiplets transforming in powers of the determinant/anti-determinant
representation of the unitary gauge groups. If all the special unitary nodes
in the U-SU quiver are balanced, the maximal unitary quiver consists of
unitary gauge nodes only.

We demonstrate this construction for the $(A_{k},A_{k})$ SCFTs, focusing on
the $k=3$ and the $k=4$ cases. For this family of SCFTs, all special unitary
nodes in the U-SU quiver are balanced. For the case $k=3$, we find a duality
of quotient realizations, while for the case $k=4$ we find a quadrality up to
some discrete identification. The maximal unitary quiver turns out to be an
Abelian quiver gauge theory with $\frac{k(k-1)}{2}$ $U(1)$ gauge nodes and
with $(k-2)$ Abelian hypermultiplets, which we write down explicitly for $k=3$
and $k=4$. In addition, the 3d mirror is shown to be a complete graph of $k+1$
vertices with edge multiplicity 1. One can check that the rank of the UV-
manifest CB symmetry (or equivalently the number of FI parameters) of the
maximal unitary quiver precisely matches the rank of the UV-manifest HB
symmetry (or equivalently the number of independent mass parameters) as
expected.

In Appendix B – Appendix D, we extend our construction to the
$D^{b}_{p}(SU(N))$ SCFTs. The circle reduction of these SCFTs leads U-SU
quivers which we review in Appendix B. In Appendix C, we determine the
subclass of SCFTs which admit hyperkähler quotient N-ality. We show that this
subclass coincides with the subclass of non-isolated SCFTs of the Argyres-
Douglas type i.e. non-isolated SCFTs with fractional scaling dimensions for
the CB operators. Every special unitary node in the U-SU quiver associated
with any SCFT of this subclass is balanced. The maximal unitary quivers are
therefore guaranteed to be unitary quiver gauge theories with Abelian
hypermultiplets. In Appendix D, we work out the maximal unitary quivers for
the $D_{6}(SU(9))$ SCFT as an explicit example, and then construct the 3d
mirror. This construction of the 3d mirror is a surprisingly simple
alternative to the Xie-Yau construction Xie and Yau (2016) that appears in the
literature Giacomelli  _et al._ (2021).

It should be evident from the above discussion that our construction can be
readily applied to trinion SCFTs built out of the $D^{b}_{p}(SU(N))$ theories,
some of which can also be realized as $(A,G)$ theories. However, unlike the
families of SCFTs studied in this paper, the U-SU quivers associated to
trinion SCFTs are generically bad theories in the Gaiotto-Witten sense, and
quotient N-ality for such theories needs a more careful analysis. It is also
tantalizing to contemplate whether generic $(G,G^{\prime})$ theories admit
quotient N-alities similar to the $(A,A^{\prime})$ theories. This will require
understanding IR N-alities in 3d SO-Sp quiver gauge theories with emergent CB
symmetry.

Acknowledgements The author would like to thank Simone Giacomelli, Amihay
Hanany, Zohar Komargodski, and Andrew Neitzke for correspondence and
discussion on related issues. The author would like to thank the organizers of
the program “Hyperkahler quotients, singularities, and quivers” at the Simons
Center for Geometry and Physics where some of the results presented in this
work were discussed. The author gratefully acknowledges the hospitality of the
Simons Summer Workshop 2023 during which part of the work was done.

## References

  * Argyres  _et al._ (1996) P. C. Argyres, M. Ronen Plesser, N. Seiberg, and E. Witten, Nucl. Phys. B461, 71 (1996), arXiv:hep-th/9511154 .
  * Argyres and Douglas (1995) P. C. Argyres and M. R. Douglas, Nucl. Phys. B448, 93 (1995), arXiv:hep-th/9505062 .
  * Klemm  _et al._ (1996) A. Klemm, W. Lerche, P. Mayr, C. Vafa, and N. P. Warner, Nucl. Phys. B 477, 746 (1996), arXiv:hep-th/9604034 .
  * Katz  _et al._ (1997) S. H. Katz, A. Klemm, and C. Vafa, Nucl. Phys. B 497, 173 (1997), arXiv:hep-th/9609239 .
  * Shapere and Vafa (1999) A. D. Shapere and C. Vafa, (1999), arXiv:hep-th/9910182 .
  * Argyres  _et al._ (2012) P. C. Argyres, K. Maruyoshi, and Y. Tachikawa, JHEP 10, 054 (2012), arXiv:1206.4700 [hep-th] .
  * Gaiotto  _et al._ (2010) D. Gaiotto, A. Neitzke, and Y. Tachikawa, Commun. Math. Phys. 294, 389 (2010), arXiv:0810.4541 [hep-th] .
  * Cecotti  _et al._ (2010) S. Cecotti, A. Neitzke, and C. Vafa, (2010), arXiv:1006.3435 [hep-th] .
  * Nanopoulos and Xie (2011) D. Nanopoulos and D. Xie, JHEP 05, 071 (2011), arXiv:1011.1911 [hep-th] .
  * Closset  _et al._ (2021) C. Closset, S. Giacomelli, S. Schafer-Nameki, and Y.-N. Wang, JHEP 05, 274 (2021), arXiv:2012.12827 [hep-th] .
  * Giacomelli  _et al._ (2021) S. Giacomelli, N. Mekareeya, and M. Sacchi, JHEP 03, 242 (2021), arXiv:2012.12852 [hep-th] .
  * Dey (2023a) A. Dey, (2023a), arXiv:2307.02525 [hep-th] .
  * Dey (2022) A. Dey, (2022), arXiv:2210.09319 [hep-th] .
  * Cecotti and Del Zotto (2013) S. Cecotti and M. Del Zotto, JHEP 01, 191 (2013), arXiv:1210.2886 [hep-th] .
  * Cecotti  _et al._ (2013) S. Cecotti, M. Del Zotto, and S. Giacomelli, JHEP 04, 153 (2013), arXiv:1303.3149 [hep-th] .
  * (16) A. Dey, In preparation .
  * Xie and Yau (2017) D. Xie and S.-T. Yau, (2017), arXiv:1701.01123 [hep-th] .
  * Buican  _et al._ (2015) M. Buican, S. Giacomelli, T. Nishinaka, and C. Papageorgakis, JHEP 02, 185 (2015), arXiv:1411.6026 [hep-th] .
  * Dey (2023b) A. Dey, (2023b), arXiv:2308.12337 [hep-th] .
  * Witten (2003) E. Witten, (2003), arXiv:hep-th/0307041 [hep-th] .
  * Dey (2021) A. Dey, JHEP 07, 199 (2021), arXiv:2004.09738 [hep-th] .
  * Xie and Yau (2016) D. Xie and S.-T. Yau, (2016), arXiv:1602.03529 [hep-th] .

##  Appendix A The Quiver Mutations

Consider a generic theory ${\cal T}$ in the class of quivers given in Figure
4. We will require that at least one of the special unitary gauge nodes in
${\cal T}$ is balanced, while the remaining ones are overbalanced. The unitary
gauge nodes are either balanced or overbalanced, and connected to any number
of Abelian hypers. One can then introduce four quiver mutations on ${\cal T}$
– shown in Figure 8 – as follows:

  1. 1.

Mutation $I$ (${\cal O}_{I}$), shown in the first row of Figure 8, involves
replacing a balanced $SU(N_{\alpha})$ node by a $U(N_{\alpha}-1)$ node and a
single Abelian hypermultiplet with the following charge vector $\bm{Q}$:

$\displaystyle\bm{Q}=\Big{(}0,\ldots,N_{\alpha_{1}},N_{\alpha_{2}},-(N_{\alpha}-1),N_{\alpha_{3}},N_{\alpha_{4}},\ldots,0\Big{)},$
(12)

where $-(N_{\alpha}-1)$ is the charge under the $U(N_{\alpha}-1)$ gauge node,
and $\\{N_{\alpha_{i}}\\}$ are the charges under those gauge nodes
$\\{U(N_{\alpha_{i}})\\}$ which are connected to $U(N_{\alpha}-1)$ by
bifundamental hypers. The hypermultiplet is not charged under any other gauge
node in the quiver.

  2. 2.

Mutation $I^{\prime}$ (${\cal O}_{I^{\prime}}$), shown in the second row of
Figure 8, acts on a $U(N_{\alpha})$ gauge node with balance parameter
$e_{\alpha}=1$ and a single Abelian hypermultiplet. Without loss of
generality, we choose the latter to have charge $N_{\alpha}$ under the
$U(N_{\alpha})$ node and charges $\\{Q_{a}\\}$ under the other unitary gauge
groups. The mutation deletes the Abelian hyper, replaces the $U(N_{\alpha})$
node with a $U(N_{\alpha}+1)$ node, and ungauges a specific $U(1)$ symmetry of
the quiver. The particular $U(1)$ symmetry to be ungauged corresponds to the
following generator:

$J_{G}\equiv\sum_{a\neq\alpha}\,\Big{(}\frac{Q_{a}}{N_{a}}\Big{)}\,J_{a}+\sum_{i}\,J_{\alpha_{i}}-J_{\alpha},$
(13)

where $J_{i}$ denotes the generator for the $\mathfrak{u}(1)$ topological
symmetry associated with the gauge group $U(N_{i})$. On the RHS, the first sum
extends over all the gauge nodes in ${\cal T}$ under which the Abelian
hypermultiplet is charged, except for the $U(N_{\alpha})$ node. The second sum
extends over all the gauge nodes which are connected to $U(N_{\alpha})$ by
bifundamental hypers. In the special case where the Abelian hyper is only
charged under the latter set of gauge nodes with charges $\\{-N_{i}\\}$, the
ungauging operation gives an $SU(N_{\alpha}+1)$. In this case, the ${\cal
O}_{I^{\prime}}$ operation therefore reduces to the inverse of the operation
${\cal O}_{I}$. For the generic case, the ungauging operation with respect to
$\mathfrak{u}(1)_{G}$ is denoted by $``\Bigg{/}U(1)"$ in the quiver.

  3. 3.

Mutation $II$ (${\cal O}_{II}$), shown in the third row of Figure 8, acts on a
$U(N_{\alpha})$ gauge node with balance parameter $e_{\alpha}=0$ and $P\geq 1$
Abelian hypermultiplets. Without loss of generality, we choose the latter to
have charge $N_{\alpha}$ under the $U(N_{\alpha})$ node and charges
$\\{Q^{l}_{a}\\}_{l=1,\ldots,P}$ under the other unitary gauge groups, such
that the charge vector has the form:

$\displaystyle\bm{Q}^{l}=(Q^{l}_{1},\ldots,Q^{l}_{\alpha_{1}},Q^{l}_{\alpha_{2}},N_{\alpha},Q^{l}_{\alpha_{3}},Q^{l}_{\alpha_{4}},\ldots,Q^{l}_{L}),$
(14)

for $l=1,\ldots,P$, $L$ being the total number of nodes in the quiver, and
$\\{N_{\alpha_{i}}\\}$ being the charges under those gauge nodes
$\\{U(N_{\alpha_{i}})\\}$ which are connected to $U(N_{\alpha})$ by
bifundamental hypers. Under this mutation, the gauge and flavor nodes remain
invariant, while the $P$ Abelian hypermultiplets in ${\cal T}$ are mapped to
$P$ Abelian hypermultiplets with the following charge vectors:

$\displaystyle\bm{Q}^{\prime
l}=(-Q^{l}_{1},\ldots,-Q^{l}_{\alpha_{1}}-N_{\alpha_{1}},-Q^{l}_{\alpha_{2}}-N_{\alpha_{2}},N_{\alpha},-Q^{l}_{\alpha_{3}}-N_{\alpha_{3}},-Q^{l}_{\alpha_{4}}-N_{\alpha_{4}},\ldots,-Q^{l}_{L}).$
(15)

One can check that this operation squares to an identity operation.

  4. 4.

Mutation $III$ (${\cal O}_{III}$), shown in the fourth row of Figure 8, acts
on a $U(N_{\alpha})$ gauge node with balance parameter $e_{\alpha}=-1$ and
$P\geq 1$ Abelian hypermultiplets. The charges of the Abelian hypermultiplets
may be chosen as in (14). The mutation replaces a $U(N_{\alpha})$ gauge node
by a $U(N_{\alpha}-1)$ node and a $U(1)$ node where the latter node has a
single fundamental hyper. In addition, the $P$ Abelian hypermultiplets get
mapped by the mutation to $P$ Abelian hypermultiplets with the following
charge vectors:

$\displaystyle\bm{Q}^{\prime
l}=\Big{(}1,Q^{l}_{1},\ldots,Q^{l}_{\alpha_{1}}+N_{\alpha_{1}},Q^{l}_{\alpha_{2}}+N_{\alpha_{2}},-(N_{\alpha}-1),Q^{l}_{\alpha_{3}}+N_{\alpha_{3}},Q^{l}_{\alpha_{4}}+N_{\alpha_{4}},\ldots,Q^{l}_{L}\Big{)},$
(16)

where the first entry denotes the charge under the new $U(1)$ node and
$-(N_{\alpha}-1)$ is the charge under the $U(N_{\alpha}-1)$ node. For the
other nodes, only the charges associated with those connected to
$U(N_{\alpha}-1)$ with bifundamental hypers get transformed under the
mutation. Finally, one can check that the composition of ${\cal
O}_{I^{\prime}}$ with ${\cal O}_{III}$, with the ${\cal O}_{I^{\prime}}$
acting on the $U(N_{\alpha}-1)$ node, gives the identity operation.

$N_{\alpha_{2}}$ $N_{\alpha}$ $N_{\alpha_{3}}$ $N_{\alpha_{1}}$ $N_{\alpha_{4}}$ $({\cal T})$ |  ${\cal O}_{I}$ |  $N_{\alpha_{2}}$ $N_{\alpha}-1$ $N_{\alpha_{3}}$ $N_{\alpha_{1}}$ $N_{\alpha_{4}}$ $({\cal T}^{\vee})$
---|---|---
$N_{\alpha_{2}}$ $N_{\alpha}$ $N_{\alpha_{3}}$ $N_{\alpha_{1}}$ $N_{\alpha_{4}}$ $({\cal T})$ |  ${\cal O}_{I^{\prime}}$ |  $N_{\alpha_{2}}$ $N_{\alpha}+1$ $N_{\alpha_{3}}$ $N_{\alpha_{1}}$ $N_{\alpha_{4}}$ $\Bigg{/}U(1)$ $({\cal T}^{\vee})$
$N_{\alpha_{2}}$ $N_{\alpha}$ $N_{\alpha_{3}}$ $N_{\alpha_{1}}$ $N_{\alpha_{4}}$ $\bm{Q}$ $({\cal T})$ |  ${\cal O}_{II}$ |  $N_{\alpha_{2}}$ $N_{\alpha}$ $N_{\alpha_{3}}$ $N_{\alpha_{1}}$ $N_{\alpha_{4}}$ $\bm{Q}^{\prime}$ $({\cal T}^{\vee})$
$N_{\alpha_{2}}$ $N_{\alpha}$ $N_{\alpha_{3}}$ $N_{\alpha_{1}}$ $N_{\alpha_{4}}$ $\bm{Q}$ $({\cal T})$ |  ${\cal O}_{III}$ |  $N_{\alpha_{2}}$ $N_{\alpha}-1$ $N_{\alpha_{3}}$ $1$ $1$ $N_{\alpha_{1}}$ $N_{\alpha_{4}}$ $(1,\bm{Q}^{\prime})$ $({\cal T}^{\vee})$
Figure 8: The four quiver mutations.

##  Appendix B Circle reduction and U-SU quivers of $D^{b}_{p}(SU(N))$ SCFTs

In this section, we review the 3d U-SU quivers that arise from the circle-
reduction of the $D^{b}_{p}(SU(N))$ theories, following the results in Closset
_et al._ (2021). The quiver data can be summarized as follows:

$[D^{b}_{p}(SU(N))]_{3d}$ : |  $N$ $n_{1}$ $n_{2}$ $n_{l-1}$ $n_{l}$ |  if $b=N$ and $\frac{p}{N}\not\in\mathbb{Z}$
---|---|---
|  $N$ $n_{1}$ $n_{2}$ $n_{l-1}$ $n_{l}$ 1 |  otherwise

The integers $\\{n_{k}\\}$ which label the various gauge nodes of the above
linear quiver are given as:

$n_{k}=\lfloor{N-\frac{k\,b}{p}}\rfloor,\qquad k\leq l,$ (17)

where $l$ is the number of gauge nodes in the linear quiver. A gauge node
labelled by an integer $n_{k}$ is a $U(n_{k})$ node if
$\frac{k\,b}{p}\not\in\mathbb{Z}$ or an $SU(n_{k})$ node
$\frac{k\,b}{p}\in\mathbb{Z}$ respectively. Finally, the length of the quiver
$l$ is given by:

$\displaystyle l=\begin{cases}&p-1-\frac{p}{N}\quad\text{if}\quad
b=N\,\text{and}\,\frac{p}{N}\in\mathbb{Z},\\\ &\lfloor
p-\frac{p}{N}\rfloor\quad\text{if}\quad
b=N\,\text{and}\,\frac{p}{N}\not\in\mathbb{Z},\\\ &p-1\quad\text{if}\quad
b=N-1.\end{cases}$ (18)

Given the above rules, one can readily check that the quiver associated to the
$D_{2}(SU(2n-1))$ SCFT is a $U(n-1)$ SQCD with $N_{f}=2n-1$. A slightly more
non-trivial example is that of the $D_{6}(SU(9))$ SCFT which we study in
Section D. In this case, the length of the quiver is $l=\lfloor
p-\frac{p}{N}\rfloor=5$, with the labels of the nodes being :
$\\{n_{k}=\lfloor 9-\frac{3k}{2}\rfloor\\}_{k=1,\ldots,5}=\\{7,6,4,3,1\\}$.
Since $\frac{k\,b}{p}=\frac{3k}{2}$, the nodes corresponding to $k=2$ and
$k=4$ are special unitary nodes while the rest are unitary nodes. This leads
to the final form of the quiver:

$[D_{6}(SU(9))]_{3d}$ : 9 7 6 4 3 1

(19)

##  Appendix C Condition for Hyperkähler Quotient N-ality: $D^{b}_{p}(SU(N))$
SCFTs

In this section, we find the subclass of $D^{b}_{p}(SU(N))$ SCFTs which have
the property of hyperkähler quotient N-ality. The general form of the 3d U-SU
quiver that arises from the circle-reduction of a $D^{b}_{p}(SU(N))$ SCFT is
given in Appendix B. A generic U-SU quiver will have an emergent CB symmetry
and admit an IR duality sequence if at least one of the SU nodes is balanced.
Therefore, our task is to find the condition involving the parameters $b,p$
and $N$ that ensures that the U-SU quiver has one or more balanced SU nodes.

We begin by looking at the larger subclass of $D^{b}_{p}(SU(N))$ SCFTs which
have SU nodes (balanced or overbalanced). Since the SU nodes are associated
with the marginal couplings, this is in fact the subclass of non-isolated
SCFTs. From Appendix B, we know that the $k$-th node in the quiver with label
$n_{k}$ is special unitary, if the integer k satisfies the condition:

$\frac{k\,b}{p}\in\mathbb{Z},\quad\text{where}\quad k\leq l,$ (20)

with $l$ being the length of quiver given in (18). The larger subclass
therefore consists of SCFTs for which the associated U-SU quiver satisfies the
above condition for at least one value of $k$. For such a $k$, one has a
$SU(n_{k})$ node with $n_{k}=\lfloor
N-\frac{k\,b}{p}\rfloor=(N-\frac{k\,b}{p})$. This node will be balanced if
$n_{k-1}+n_{k+1}=2n_{k}-1$ which translates to the condition:

$\lfloor n_{k}+\frac{b}{p}\rfloor+\lfloor n_{k}-\frac{b}{p}\rfloor=2n_{k}-1.$
(21)

This condition is satisfied for a generic $k$ satisfying (20) if the
parameters $b$ and $p$ obey the constraint :

$\frac{b}{p}\not\in\mathbb{Z}.$ (22)

Note that this condition is independent of $k$. This implies that for a given
theory in the subclass of non-isolated $D^{b}_{p}(SU(N))$ SCFTs, every SU node
in the 3d U-SU quiver will be balanced provided the above condition is
satisfied. Following the discussion in Section III, any such quiver will admit
a duality sequence leading to maximal unitary quivers which only consist of
unitary gauge nodes (since all SU nodes are balanced and therefore admit
mutation $I$) and a total of $GCD(b,p)-1$ Abelian hypermultiplets – one each
from the $GCD(b,p)-1$ balanced special unitary nodes, in addition to
fundamental/bifundamental hypers.

Let us investigate in a bit more detail what the condition (22) really means.
In the Type IIB picture, the deformed singularity for the $D^{b}_{p}(SU(N))$
has the following form:

$\widehat{F}(\bm{x},z)=F(\bm{x},z)+\sum_{\begin{subarray}{c}0\leq i\leq p-1\\\
0\leq j\leq N-2\end{subarray}}\,u_{ij}\,x^{j}_{3}\,z^{i}=0,$ (23)

where the polynomial $F(\bm{x},z)$ has the form given in (5), and the
coefficients $\\{u_{ij}\\}$ have the scaling dimensions:

$\Big{[}u_{ij}\Big{]}=\Big{(}N-j-i\,\frac{b}{p}\Big{)}.$ (24)

For $\frac{b}{p}\in\mathbb{Z}$, none of the CB operators can have fractional
scaling dimensions, while for $\frac{b}{p}\in\mathbb{Z}$ the SCFT generically
has CB operators with fractional scaling dimensions and are of the Argyres-
Douglas type. Therefore, we conclude that the subclass of $D^{b}_{p}(SU(N))$
SCFTs which admit hyperkähler quotient N-ality in the sense discussed in
Section III is given by the subclass of non-isolated SCFTs of the Argyres-
Douglas type.

##  Appendix D Illustrative Example : Maximal Unitary Quivers and the 3d
Mirror of $D_{6}(SU(9))$

In this section, we will present a concrete example where we determine the
maximal unitary quiver and the 3d mirror of a $D^{b}_{p}(SU(N))$ SCFT which
admits hyperkähler quotient N-ality. We choose the SCFT $D_{6}(SU(9))$ which
has a conformal manifold of dimension 2 corresponding to two special unitary
vector multiplets in the 3d U-SU quiver which is given in (19).

9 7 6 4 3 1 $({\cal T})$ |  ${\cal O}_{I}$ |  $(7,-5,4)$ 9 7 5 4 2 1 $({\cal T}_{1})$
---|---|---
|  |  ${\cal O}_{I}$
$(1,7,-3,2)$ $(1,5,-3,1)$ 9 7 5 3 2 1 1 1 $({\cal T}_{\rm maximal})$ |  ${\cal O}_{III}$ |  $(7,-5,4)$ $(4,-2,1)$ 9 7 5 4 2 1 $({\cal T}_{2})$
Figure 9: Derivation of ${\cal T}_{\rm maximal}$ for the $D_{6}(SU(9))$ SCFT.

For the sake of brevity, we will not list the complete list of quivers in the
N-al set, although they can be readily determined following the general
prescription presented in Section III. Instead, we will discuss a
(sub)sequence of IR dualities that leads to the subset of maximal unitary
quivers. The sequence of dualities is shown in Figure 9. We begin by
implementing mutation $I$ at the balanced $SU(6)$ node, followed by another
mutation $I$ at the balanced $SU(4)$ node. Each mutation introduces an Abelian
hypermultiplet with charges as shown, leading to the quiver ${\cal T}_{2}$
which has only unitary nodes. The $U(4)$ node in ${\cal T}_{2}$ has balance
parameter $e=-1$ and therefore admits a mutation $III$. The resultant quiver
(on the bottom left corner) has only unitary gauges nodes which are either
balanced or overbalanced and do not admit another mutation $III$. Therefore,
one can identify this quiver as a maximal unitary quiver ${\cal T}_{\rm
maximal}$. The subset of maximal unitary quivers may be obtained from this
quiver by enumerating the distinct Lagrangians that arise from implementing a
sequence of mutation $II$ at the balanced nodes.

Note that $\text{rk}(\mathfrak{g}^{\rm UV}_{\rm C}({\cal T}))=3$ which implies
that one can only turn on two FI parameters for the quiver, and the rank
increases along the duality sequence by 1 at each step. For the maximal
unitary quiver, the number of FI parameters is maximized with
$\text{rk}(\mathfrak{g}^{\rm UV}_{\rm C}({\cal T}_{\rm maximal}))=6$.

9 7 5 3 2 1 1 1 1 1 1 $({\cal T}_{\rm good}\oplus{\cal T}^{(1)}_{\rm decoupled}\oplus{\cal T}^{(2)}_{\rm decoupled})$ |  $S_{1}$ |  $(1,7,-3,2)$ 9 7 5 3 2 1 1 1 1 1 $({\cal T}^{(1)}_{\rm good}\oplus{\cal T}^{(2)}_{\rm decoupled})$ |  $S_{2}$ |  $(1,7,-3,2)$ $(1,5,-3,1)$ 9 7 5 3 2 1 1 1 $({\cal T}_{\rm maximal})$
---|---|---|---|---
$T(U(6))$ 5 7 1 2 1 2 1 1 1 2 $(\widetilde{{\cal T}}_{\rm good}\oplus\widetilde{{\cal T}}^{(1)}_{\rm decoupled}\oplus\widetilde{{\cal T}}^{(2)}_{\rm decoupled})$ |  $S_{1}$ |  $T(U(6))$ 5 7 1 1 1 1 1 1 1 1 1 $(\widetilde{{\cal T}}^{(1)}_{\rm good}\oplus\widetilde{{\cal T}}^{(2)}_{\rm decoupled})$ |  $S_{2}$ |  $T(U(6))$ 1 5 7 1 1 1 1 1 $(\widetilde{{\cal T}}_{\rm maximal})$
Figure 10: Construction of the 3d mirror for $D_{6}(SU(9))$. The gray node
labelled denotes a $T(U(6))$ quiver tail.

Next, let us construct the 3d mirror for $D_{6}(SU(9))$. The maximal unitary
quiver ${\cal T}_{\rm maximal}$ can again be thought of as a linear quiver
gauge theory decorated with Abelian hypermultiplets attached in turn to
Abelian quiver tails. The linear quiver ${\cal T}_{\rm good}$, obtained by
stripping off the Abelian hypermultiplets plus the quiver tails, and its 3d
mirror $\widetilde{{\cal T}}_{\rm good}$ are given as:

9 7 5 3 2 1 $({\cal T}_{\rm good})$ |  |  $T(U(6))$ 5 7 3 3 $(\widetilde{{\cal T}}_{\rm good})$
---|---|---

The gray node denotes a $T(U(6))$ tail :  6 5 4 3 2 1 . Following the
prescription in Section V, one can then proceed with implementing
$S$-operations on the quiver ${\cal T}_{\rm good}$, as shown in the top row of
Figure 10, to construct the quiver ${\cal T}_{\rm maximal}$. The operations on
the mirror side are shown in the bottom row and lead to the 3d mirror of the
$D_{6}(SU(9))$ SCFT:

$T(U(6))$ 1 5 7 1 1 1 1 1 |  |  $T(U(6))$ 1 5 7 1 1
---|---|---

Given the 3d mirror quiver, one can readily check that the number of
independent mass parameters is given by $\text{rk}(\mathfrak{g}^{\rm UV}_{\rm
H}(\widetilde{{\cal T}}_{\rm maximal}))=6$, which exactly matches the number
of FI parameters in the quiver ${\cal T}_{\rm maximal}$.