bbinstitutetext: Department of Physics and Astronomy, Uppsala University, Box
516, SE-75120 Uppsala, Swedenaainstitutetext: Dipartimento di Fisica,
Università degli studi di Milano–Bicocca, and INFN, Sezione di Milano–Bicocca,
Piazza della Scienza 3, I-20126 Milano, Italy

# Phases of $\mathcal{N}=1$ Quivers in $2+1$ Dimensions

Vladimir Bashmakov b Nicola Gorini<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider the IR phases of two-node quiver theories with $\mathcal{N}\,=\,1$
supersymmetry in $d\,=\,2\,+\,1$ dimensions. It turns out that the discussion
splits into two main cases, depending on whether the Chern-Simons levels
associated with two nodes have the same sign, or the opposite signs, with the
latter case being more non-trivial. The determination of the phase diagrams
allows us to conjecture certain infrared dualities involving either two quiver
theories, or a quiver and adjoint QCD. We also provide a short discussion of
quivers possessing time reversal symmetry.

††preprint: UUITP-45/21

##  1 Introduction

Three-dimensional quantum field theories with $\mathcal{N}=1$
supersymmetry111$\mathcal{N}=1$ supersymmetric theories in three dimensions
have two real supercharges, forming a Majorana spinor. constitute a remarkable
bridge between theories with $\mathcal{N}=2$ supersymmetry, which are quite
well understood thanks to holomorphy, and genuine non-supersymmetric theories,
whose dynamics is a challenging subject. Given that we do not have neither
non-renormalization theorems, nor localization techniques at our disposal, it
may appear that $\mathcal{N}=1$ supersymmetry does not give any advantage
compared to cases without supersymmetry. Nevertheless, in the recent years our
understanding of these theories has overcome a new twist Gomis:2017ixy ;
Bashmakov:2018wts ; Benini:2018umh ; Gaiotto:2018yjh ; Benini:2018bhk ;
Choi:2018ohn ; Bashmakov:2018ghn ; Aharony:2019mbc ; Bashmakov:2019myq  (see
also Gates:1983nr ; Ohta:1997fr ; Kitao:1998mf ; Witten:1999ds ;
Bergman:1999na ; Gremm:1999su ; Acharya:2001dz ; Schwarz:2004yj ; Gukov:2002es
; Forcella:2009jj ; Amariti:2014ewa  for earlier considerations).

In particular, new tools for studying phase diagrams of $3d$ $\mathcal{N}=1$
theories were introduced and applied to the analysis of infrared (IR) dynamics
of a vector multiplet coupled to matter in the adjoint or fundamental
representations Bashmakov:2018ghn ; Choi:2018ohn . Among various interesting
phenomena observed in those examples, we mention the existence of walls in the
parameter space where the Witten index jumps, as well as the presence of
second-order (or higher) phase transition points. Both features seem to be
ubiquitous for $3d$ $\mathcal{N}=1$ theories and stem from the fact that the
theories possess real parameters.222$\mathcal{N}=2$ theories in three
dimensions also have real parameters, which allow phase transition, but do not
allow jumps of the Witten index Intriligator:2013lca .

In this paper we initiate the study of phase diagrams of $3d$ $\mathcal{N}=1$
theories with the gauge group given by a product of several non-Abelian
factors and bi-fundamental matter coupled to it, known as quiver gauge
theories. Such theories may arise as worldvolume theories of BPS domain walls
and interfaces in four-dimensional supersymmetric theories Acharya:2001dz ;
Gaiotto:2017tne ; Bashmakov:2018ghn , as well as world-volume theories of
M2/D2 branes on backgrounds with four preserved supercharges (e.g. M2 branes
probing $Spin(7)$ cones Forcella:2009jj ). There exist extensive literature on
$\mathcal{N}=2$ quiver theories (for a recent discussion see e.g.
Benini:2011mf ; Amariti:2014lla ; Benvenuti:2016wet ; Amariti:2017gsm ;
Benvenuti:2017kud ; Benvenuti:2017bpg ; Nedelin:2017nsb ; Aprile:2018oau ;
Amariti:2019pky ; Pasquetti:2019uop ; Pasquetti:2019tix ; Jain:2019lqb ;
Benvenuti:2020gvy ; Benvenuti:2020wpc  and references therein) and also some
results regarding non-supersymmetric quiver theories Karch:2016aux ;
Jensen:2017dso ; Aitken:2019mtq . For a prior studies of $\mathcal{N}=1$
quivers instead see Amariti:2014ewa .

In the present work we restrict ourselves to the two-node quivers with $SU(2)$
or $U(2)$ gauge groups and with one bi-fundamental matter multiplet. This
simple setup allows for a detailed treatment and is still rich enough to
accommodate a variety of interesting phenomena. In particular, we reveal a
rather non-trivial phase structure of these theories and identify a collection
of $\mathcal{N}=1$ superconformal field theories (SCFT). Basing on our
understanding of phase diagrams, we are able to conjecture certain infrared
dualities, some of which can be understood as the dualization of a node of a
quiver, and some others as the confinement of a node. Finally, when the gauge
groups of a two-node quiver are the same, and the Chern-Simons levels are
chosen to be opposite, the theory turns out to be time-reversal invariant.
This symmetry gives rise to certain non-renormalization theorems
Gaiotto:2018yjh  which put strong restrictions on the form of the effective
superpotential.

To get some preliminary ideas about the features of the phases of quiver
theories we are going to study, let us consider the $SU(2)_{k_{1}}\times
SU(2)_{k_{2}}$ theory with a bi-fundamental multiplet $\Phi$ (we assume for a
moment that $k_{1},k_{2}>0$). This theory has two discrete dimensionless
parameters, the Chern-Simons levels $k_{1}$ and $k_{2}$, and three continuous
dimensionful parameters, the two gauge couplings $g_{1},g_{2}$ and the matter
mass $m$. We generically expect to consider a two-dimensional phase diagram
since the number of independent dimensionless combinations we can construct in
our setup is exactly two.333More explicitly, one of the parameters is going to
be $\frac{g_{1}}{g_{2}}$, and the other is a ratio of the mass and some
combination of the couplings, e.g. $\frac{m}{g_{1}^{2}+g_{2}^{2}}$.

In the large mass limit, $|m|\gg g_{1,2}^{2}$, matter can be integrated out
semiclassically. If $m>0$, we then get at low energies

$\mathcal{N}=1\qquad SU(2)_{k_{1}+1}\times SU(2)_{k_{2}+1},$ (1)

where the one-loop renormalization of Chern-Simons levels is taken into
account (see Section 2 for some details). This theory further flows to a
purely topological CS theory Witten:1999ds ,

$SU(2)_{k_{1}}\times SU(2)_{k_{2}}\quad\text{TQFT}.$ (2)

If instead $m<0$, we get

$\mathcal{N}=1\qquad SU(2)_{k_{1}-1}\times SU(2)_{k_{2}-1},$ (3)

which can either flow to a CS theory, or break SUSY and gives rise to a
Goldstino 444Eq. 3 may suggest that when both vector multiplets break
supersymmetry, two Goldstinos arise. A more natural expectation is that we get
one massless Majorana fermion (the Goldstino) and one very light Majorana
fermion, with the mass suppressed by the scale of $m$. with a decoupled
topological sector. We note in particular that the Witten indices differ in
these two limits, which indicates the existence of a wall that separates the
two large mass limits (see Section 2 for some details).

In order to understand the transition between the two asymptotic phases, it is
useful to consider the behaviour of the theory near a line (one dimensionless
parameter involving the mass is fixed while the other is left free) where the
matter is massless and the theory has a classical moduli space of vacua. In
fact, it is natural to assume that new vacua appear near this line to
compensate the jump of the Witten index; this is going to be the wall
mentioned above.

To describe more explicitly the classical moduli space, let us note that every
complex matrix $M$ possesses a singular value decomposition of the form
$M\,=\,U\,\Sigma\,V^{\dagger}$, where $U$ and $V$ are unitary, and $\Sigma$ is
diagonal with real non-negative entries (the matrices can be chosen such that
the diagonal entries appear in descending order). Thus, we can apply two gauge
transformations, one for each factor of the gauge group, to put the matrix
$\Phi$ in a diagonal form with real diagonal entries all multiplied by an
overall phase. In total, the classical moduli space for the $SU(2)\times
SU(2)$ case is therefore given by

$\mathcal{M}=\mathbb{S}^{1}\,\times\,\mathbb{R}^{2}\,/\,S_{2}.$ (4)

whereas for the $SU(2)\times U(2)$ case is instead given by

$\mathcal{M}=\mathbb{R}^{2}\,/\,S_{2}.$ (5)

The effective superpotential is expected to receive radiative corrections
causing the classical moduli spaces to lift at the quantum level.

The rest of the paper is organised in the following way. In Section 2 we
review some specific aspects of $\mathcal{N}=1$ gauge theories with different
matter content, which are essential for the rest of the paper, including the
IR behaviour of a pure $\mathcal{N}=1$ vector multiplet, a vector multiplet
coupled to matter in the fundamental representation, and a vector multiplet
coupled to matter in the adjoint representation. In Section 3 we compute the
effective superpotential for the $SU(2)\times SU(2)$ class of quivers and
study its behaviour in some physiscally relevant asymptotic limits. This
result will allow us to study the phase diagrams of these models which we will
present in Section 4. We then proceed with the study of the phase diagrams for
$SU(2)\times U(2)$ models in Section 5. For this case we do not have the
explicit effective potentials, so we try to do some reasonably looking
assumptions about their vacuum structure close to the wall. In Section 6 we
propose dualities involving quiver theories, as well as the adjoint SQCD. In
section 7 we make some comments regarding the dynamics of quiver theories
enjoin time reversal invariance. Section 8 contains conclusions and future
directions.

##  2 Review of previous results

###  2.1 Chern-Simons levels

Since we are going to deal with Chern-Simons terms, we start with stating the
related conventions. In three dimensions, a Yang-Mills theory admits a gauge
invariant deformation given by a Chern-Simons term. This is a priori
caracterized by its level $k_{bare}\in\mathbb{Z}$. Instead of characterizing
it directly by $k_{bare}$, we prefer following the common practice and
introduce the level $k$ related to the bare one by

$k\,=\,k_{bare}-\tfrac{1}{2}\sum_{f}T(R),$ (6)

where the sum is taken over fermions in the representation $R$ of the given
gauge group and $T(R)$ is the Dynkin index555We recall to the reader that for
$SU(N)$ we have $T(F)=\frac{1}{2}$ and $T(A)=\frac{N}{2}$ for respectively the
fundamental and the adjoint representations. of the real representation $R$.
Notice that the level $k$ defined above transforms as $k\rightarrow-k$ under
time-reversal symmetry.

When a Majorana fermion of mass $m$ in a real representation $R$ is integrated
out, the level gets shifted as follows666In the case of fermions in a complex
or a pseudo-real representation, the $\tfrac{1}{2}$ factor has to be dropped
from both equations (6) and (7).

$k\rightarrow k+\tfrac{1}{2}\text{sign}(m)T(R).$ (7)

When all fermions are integrated out, the resulting level is sometimes
referred to as the infrared level. Gauge invariance requires that the bare
level $k_{bare}$ and the infrared level $k_{IR}$ are integrally quantized. At
the same time, the level $k$ used to label a theory may be either integrally
quantized or half-integrally quantized, depending on the number of fermions in
the theory and their representations (charges).

###  2.2 Level-rank duality

Theories with $U(N)$ gauge groups may have two Chern-Simons terms, the single
trace and the double-trace ones. According to the standard definition, we have

$U(N)_{k_{1},k_{2}}\,=\,\frac{SU(N)_{k_{1}}\times
U(1)_{Nk_{2}}}{\mathbb{Z}_{N}},$ (8)

and consistency requires that $k_{2}=k_{1}\ \text{mod}\ N$. We note that while
in general time reversal transformation maps a TQFT to a different TQFT, there
are some which turn out to be $T$-invariant, with an example given by
$U(N)_{N,2N}$ theories. It thus happens that different Chern-Simons theories
may in fact describe the same TQFT. An important example of this phenomenon,
used in what follows, is given by Level-Rank duality (see Hsin:2016blu  for a
modern discussion). The dual pair relevant for us is

$U(N)_{k,k}\longleftrightarrow SU(k)_{-N}.$ (9)

###  2.3 $SU(N)_{k}$ Vector multiplet

$\mathcal{N}=1$ $SU(N)_{k}$ vector multiplet, consisting of a gauge field $A$
and a Majorana fermion $\lambda$ in the adjoint representation, was first
studied in Witten:1999ds . For $k\geq N/2$ the theory flows to the
$SU(N)_{k-\frac{N}{2}}$ Chern-Simons theory which is topological. For large
values of the level, i.e. $k\gg 1$, this can be understood semiclassically by
integrating out the gaugino, whose mass is
$m_{gaugino}=-\frac{k_{bare}g^{2}}{2\pi}$. This gives the $-\frac{N}{2}$ shift
of the Chern-Simons level.
In the resulting Yang-Mills-Chern-Simons theory, the propagating modes are
massive Deser:1981wh ; Deser:1982vy  and at low energies only topological
degrees of freedom survive. The Witten index in this regime is given (up to an
overall sign777As a general comment, the sign depends on the number of
Majorana fermions with negative mass in the theory, which we can call $n_{F}$.
The overall sign is then given by $(-1)^{n_{F}}$) by

$I\,=\,\frac{1}{(N-1)!}\left(k\,-\,\frac{N}{2}\,+\,1\right)\left(k\,-\,\frac{N}{2}\,+\,2\right)...\left(k\,+\,\frac{N}{2}\,-\,1\right),$
(10)

which is just the partition function of the low-energy Chern-Simons TQFT on
the torus.888Equivalently, it can be computed by counting the number of
inequivalent Wilson lines of the theory.

For $0\leq k<N/2$ the Witten index vanishes, and it was conjectured in
Witten:1999ds  that supersymmetry is spontaneously broken in such models, thus
massless Goldstino emerges in the IR limit. This conjecture found significant
support in the results of Gomis:2017ixy , where it was claimed that, in
addition to the Goldstino $G_{\alpha}$, there must be topological degrees of
freedom described by a $U\left(\frac{N}{2}-k\right)_{\frac{N}{2}+k,N}$ Chern-
Simons theory. 999The fact that the Goldstino alone is not enough to describe
the IR physics follows from the following observations: at any value of the
level $k$ there is a 1-form symmetry in the UV moreover, for $k=0$, the UV
theory enjoys the time reversal symmetry. There are ’t Hooft anomalies
associated with both symmetries that cannot be matched by the Goldstino only
and some other d.o.f. are required. This TQFT has a non-perturbative origin,
and its precise form follows from certain dualities.

###  2.4 Vector multiplet coupled to an adjoined matter multiplet

Matter multiplet consists of a real scalar and a Majorana fermion, and
$\mathcal{N}=1$ supersymmetry allows real superpotential for matter; in what
follows we will be interested in the tree-level superpotentials given only by
the mass term. The dynamics of an $SU(N)$ vector multiplet coupled to a matter
multiplet in the adjoint representation as a function of the matter mass
parameter was described in details in Bashmakov:2018wts . Here we review the
case of $SU(2)$ gauge group, relevant for the rest of the paper.

For $k\geq 2$ and $m\gg 0$, we can integrate out the matter multiplet and get
the pure $SU(2)_{k+1}$ vector multiplet. In this limit the theory flows to a
supersymmetric vacuum hosting the $SU(2)_{k}$ CS theory. When we consider
$m\ll 0$ instead, we get the pure $SU(2)_{k-1}$ vector multiplet, which flows
to a supersymmetric vacuum with $SU(2)_{k-2}$ CS theory. Thus, there are two
asymptotic phases with different Witten indices. The transition between these
two phases happens in two stages. At $m=0$ the effective potential develops a
flat direction, and for small and positive mass a new supersymmetric vacuum
supporting the $U(1)_{2k}$ CS theory comes in from infinity of the field
space: this new vacuum compensates the jump of the Witten index. At some
finite mass value two vacua merge through a second order phase transition and
produce a single vacuum visible at large positive masses.

For $k=1$ and large positive mass the picture is not much different from the
previous case, indeed we get a single supersymmetric vacuum supporting the
$SU(2)_{1}$ TQFT. On the contrary, when the mass is large and negative, by
integrating out matter we get the $SU(2)_{0}$ vector multiplet which breaks
supersymmetry. The theory in the IR limit is then given by101010We warn the
reader that the $U(1)_{2}$ here is of the origin discussed at the end of
Section 2.3.

$G_{\alpha}\,+\,U(1)_{2}.$ (11)

At $m=0$ the effective potential develops the flat direction, and for small
positive mass we find one supersymmetric vacuum with $U(1)_{2}$ TQFT, which
came in from infinity of the field space. Using the fact that $U(1)_{2}$ is
time-reversal invariant and applying the Level-Rank duality (cf. Hsin:2016blu
),

$U(1)_{2}\longleftrightarrow
U(1)_{-2}\overset{\text{L-R}}{\longleftrightarrow}SU(2)_{1},$ (12)

we recognise the same vacuum as we have seen at the large positive mass.
Therefore, we do not have a phase transition in this case. We note that at
positive and small mass values, the non-supersymmetric vacuum coexists with
the supersymmetric one, and so the former is meta-stable.

Finally, for $k=0$, in the $m=0$ point we get the supersymmetry enhancement to
$\mathcal{N}=2$, and the theory shows a runaway behaviour Affleck:1982as . For
non-zero masses the runaway behaviour stabilizes and the superpotential shows
a single trivial supersymmetric vacuum for both positive and negative masses.

###  2.5 Vector multiplet coupled to fundamental matter multiplets

In this section we review the physics of $SU(2)$ and $U(2)$ vector multiplets
coupled to $F$ matter multiplets in the fundamental representation as a
function of the matter mass $m$ (we assume that all flavours are given the
same mass). This is a particular case of the results in Choi:2018ohn .

Consider first $U(2)_{k+1,k}$ theories with $F$ fundamentals.

  * •

$F=1$: At large positive mass, by integrating out the matter, we get a single
vacuum with $U(2)_{k+\frac{3}{2},k+\frac{1}{2}}$ vector multiplet. For large
and negative mass we get a single vacuum with
$U(2)_{k+\frac{1}{2},k-\frac{1}{2}}$ vector multiplet. In the intermediate
phase $0<m<m_{*}$ (where $m_{*}$ is some positive value) in addition to the
vacuum seen at negative masses, a new vacuum with $U(1)_{k+\tfrac{1}{2}}$
vector multiplet appears. At $m=0$ there is a wall, and $m_{*}$ is a phase
transition point.

  * •

$F\geq 2$: At large positive masses, we get a single vacuum with
$U(2)_{k+\tfrac{2+F}{2},k+\tfrac{F}{2}}$ vector multiplet. For large and
negative masses we get a single vacuum with
$U(2)_{k+\tfrac{2-F}{2},k-\tfrac{F}{2}}$ vector multiplet. In the intermediate
phase $0<m<m_{*}$ in addition to the negative masses vacuum, two new vacua
appear. One of them hosts the $\frac{U(F)}{U(F-1)\times U(1)}$ $\mathcal{N}=1$
non-linear sigma model (coming from spontaneously broken $SU(F)$ global
symmetry) together with the decoupled $U(1)_{k}$ vector multiplet. The second
new vacuum hosts the $\frac{U(F)}{U(F-2)\times U(2)}$ $\mathcal{N}=1$ NLSM.
Again, at $m=0$ there is a wall, and at $m=m_{*}$ is a phase transition point.

A peculiarity of this type of models is that, in contrast to expectations, the
three vacua living in the intermediate phase merge in just a single phase
transition instead of two distinct ones. This phenomenon was dubbed
supercriticality.

Next we discuss $SU(2)_{k+\tfrac{F}{2}+1}$ theories with $F$ fundamentals. It
worth reminding that the global symmetry here is not just $U(F)$, but rather
$Sp(F)$, thanks to the pseudo-real nature of the $SU(2)$ fundamental
representation. For large and positive mass we get a single vacuum with
$SU(2)_{k+F+1}$ vector multiplet, while for negative masses we have
$SU(2)_{k+1}$ vector multiplet. In the intermediate phase with $0<m<m_{*}$, in
addition to the negative mass vacuum, a new vacuum come in from infinity and
supports the $\frac{Sp(F)}{Sp(1)\times Sp(F-1)}$ $\mathcal{N}=1$ NLSM.

The authors of Choi:2018ohn  described a family of dualities between such SQCD
theories, among which there is the following 111111V.B. thanks Adar Sharon for
a discussion of this duality.

$U(2)_{2,1}+2\Phi\longleftrightarrow SU(2)_{-2}+2\tilde{\Phi}.$ (13)

This duality is quite subtle and, if correct, has quite far-reaching
consequences, such as symmetry enhancement on the $U(2)$ side from
$SU(2)\times U(1)$ to $Sp(2)$ at the IR fixed point. A puzzling aspect about
this duality is the mismatch of the phases (described above), when the
theories are deformed away from the fixed point by the mass deformation. In
particular, while the $U(2)$ theory has three vacua in the intermediate phase,
the the $SU(2)$ theory has just two of them. A possible explanation of this
apparent mismatch is the breaking of supercriticality argumentation due to
e.g. non-perturbative corrections. Giving up supercriticality, one can imagine
the intermediate phase with two vacua, matching exactly the $SU(2)$ vacua.

${\ast}\,{\ast}\,{\ast}$

While it is easy to get the vacua for large mass values, it is more subtle to
understand the dynamics in the intermediate phase for which one needs to know
the effective superpotential exactly. For this reason we turn to the
evaluation of the effective superpotential for the two-node quiver theories as
described in the introduction.

##  3 Effective superpotential for $SU(2)\times SU(2)$ theories

In this section we consider two-node quiver theories with $SU(2)_{k_{1}}\times
SU(2)_{k_{2}}$ gauge group and with one bi-fundamental matter multiplet and
compute the effective superpotential, the superspace analogue of the effective
potential discussed first by Coleman and Weinberg Coleman:1973jx . It turns
out to be useful to use supergraph formalism for this computation, in terms of
which the leading contribution comes at 1-loop order.121212One can work in
components as well, but then the two-loop computation would be needed, since
the one-loop result vanishes. We follow closely the discussion of Choi:2018ohn
, a detailed discussion of the $\mathcal{N}=1$ superspace and supergraph
formalism can be found in Gates:1983nr . In fact, up to some point, the
discussion can be held at a more general level, therefore we will start by
considering the case of $G_{1}\times G_{2}$ gauge groups and then specialize
to the case of $SU(2)\times SU(2)$.

Figure 1: Feynman Supergraphs contributing to the 1-loop effective
Superpotential.

The effective superpotential is of the form Choi:2018ohn

$\mathcal{W}_{\text{eff}}\,=\,\int\frac{d^{3}p}{(2\pi)^{3}}d^{2}\theta^{\prime}\,\delta(\theta-\theta^{\prime})\Sigma(p,\theta^{\prime})\delta(\theta^{\prime}-\theta),$
(14)

where $\Sigma$ at the one-loop order is given by the sum of the diagrams in
Figure 1.
We note that the form of the series expansion is exactly the same as can be
found in Coleman:1973jx , albeit with Feynman rules replaced by their
superspace counterpart. We thus state the relevant superspace Feynman rules
now.

Let $\Gamma_{\alpha}^{A},\,\hat{\Gamma}_{\alpha}^{M}$ be vector multiplet
superfields of the first and the second gauge groups respectively; here $A,M$
are colour indices and $\alpha$ is the spinor index. The two gauge propagators
are (147)

$\displaystyle\braket{\Gamma_{\alpha}^{A}(p)\Gamma^{B,\beta}(-p)}$
$\displaystyle=\delta^{AB}\
\frac{\delta_{\alpha}^{\beta}(\kappa_{1}\,D^{2}\,+\,p^{2})\,+\,(\kappa_{1}\,-\,D^{2})\,{p_{\alpha}}^{\beta}}{p^{2}\,(\kappa_{1}^{2}\,+\,p^{2})}\equiv\delta^{AB}{(\Delta_{1})_{\alpha}}^{\beta}$
(15)
$\displaystyle\braket{\hat{\Gamma}_{\alpha}^{M}(p)\hat{\Gamma}^{N,\beta}(-p)}$
$\displaystyle=\delta^{MN}\
\frac{\delta_{\alpha}^{\beta}(\kappa_{2}\,D^{2}\,+\,p^{2})\,+\,(\kappa_{2}\,-\,D^{2})\,{p_{\alpha}}^{\beta}}{p^{2}\,(\kappa_{2}^{2}\,+\,p^{2})}\equiv\delta^{MN}{(\Delta_{2})_{\alpha}}^{\beta}$
(16)

where $\kappa_{i}=\frac{kg_{i}^{2}}{2\pi}$.
There are also three types of vertices joining two gauge fields and two matter
fields131313As usually, there also exist a cubic vertex with two matter legs
and one gauge boson leg but they do not contribute since, in the Landau gauge,
the gauge propagators are transverse, namely $(\Delta_{1,2})_{\alpha}^{\
\beta}D_{\beta}=0$.:
we get the rules

$\displaystyle\Gamma^{(4)}_{\Gamma\Gamma\bar{\phi}\phi}$
$\displaystyle=\braket{\Gamma_{\alpha}^{A}(p)\Gamma^{B,\beta}(q){\Phi_{i}}^{\hat{j}}(r){\bar{\Phi}_{\hat{k}}}^{\
l}(-p-q-r)}\,=\,-\frac{g_{1}^{2}}{2}(T^{(A})_{i}^{\ n}(T^{B)})_{n}^{\
l}\,{\delta^{\hat{j}}}_{\hat{k}}\,{\delta_{\alpha}}^{\beta}$ (17)
$\displaystyle\Gamma^{(4)}_{\hat{\Gamma}\hat{\Gamma}\bar{\phi}\phi}$
$\displaystyle=\braket{\hat{\Gamma}_{\alpha}^{M}(p)\hat{\Gamma}^{N,\beta}(q){\Phi_{i}}^{\hat{j}}(r){\bar{\Phi}_{\hat{k}}}^{\
l}(-p-q-r)}\,=\,-\frac{g_{2}^{2}}{2}(K^{(M})_{\hat{k}}^{\
\hat{n}}(K^{N)})_{\hat{n}}^{\
\hat{j}}\,{\delta_{i}}^{\l}\,{\delta_{\alpha}}^{\beta},$ (18)
$\displaystyle\Gamma^{(4)}_{\Gamma\hat{\Gamma}\bar{\phi}\phi}$
$\displaystyle=\braket{\Gamma_{\alpha}^{A}(p)\hat{\Gamma}^{M,\beta}(q){\Phi_{i}}^{\hat{j}}(r){\bar{\Phi}_{\hat{k}}}^{\
l}(-p-q-r)}=\frac{1}{2}g_{1}g_{2}(T^{A})_{i}^{\ l}(K^{M})_{\hat{k}}^{\
\hat{j}}\,{\delta_{\alpha}}^{\beta}.$ (19)

Above unhatted lower case letters $i,j,...$ are fundamental indices and $A,B$
are adjoint indices of $G_{1}$, with $T^{A}$ being its generators. Similarly,
hatted lower case letters are fundamental indices and $M,N$ are adjoint
indices of $G_{2}$, with $K^{A}$ being its generators.

We can now proceed with the computation of $\Sigma$. For this purpose, we find
useful to introduce the following matrices

$\displaystyle M^{AB}$
$\displaystyle=g^{2}_{1}\text{Tr}\,\bar{\Phi}T^{(A}T^{B)}\Phi$ (20a)
$\displaystyle N^{MN}$ $\displaystyle=g^{2}_{2}\text{Tr}\,\Phi
K^{(M}K^{N)}\bar{\Phi}$ (20b) $\displaystyle G^{AM}$
$\displaystyle=-g_{1}g_{2}\text{Tr}\,\bar{\Phi}T^{A}\Phi K^{M}.$ (20c)

which allow us a more compact treatment of the computation. We see indeed that
$\Sigma$ is given, at the leading order, by the sum of the one-loop
contributions corresponding to Feynman diagrams appearing in Figure 1

$\displaystyle\Sigma_{\text{1-loop}}$
$\displaystyle=-\tfrac{1}{2}\,\text{Tr}\,M\,\Delta_{1}\,-\,\tfrac{1}{2}\,\text{Tr}\,N\,\Delta_{2}\,+$
(21) $\displaystyle\
\quad+\tfrac{1}{4}\text{Tr}\,(M\,\Delta_{1})^{2}\,+\,\tfrac{1}{2}\text{Tr}\,(G)^{\dagger}\,\Delta_{1}\,G\,\Delta_{2}+\,\tfrac{1}{4}\text{Tr}\,(N\,\Delta_{2})^{2}\,-$
$\displaystyle\
\quad-\tfrac{1}{6}\text{Tr}\,(M\,\Delta_{1})^{3}\,-\,\tfrac{1}{2}\text{Tr}\,G^{\dagger}\,\Delta_{1}\,M\,\Delta_{1}\,G\,\Delta_{2}\,-\,\tfrac{1}{2}\text{Tr}\,G\,\Delta_{2}\,N\,\Delta_{2}\,G^{\dagger}\,\Delta_{1}\,-\,\tfrac{1}{6}\text{Tr}\,(N\,\Delta_{2})^{3}\,+\,...$

The trace in this expression is taken both over spinor and colour indices (for
each gauge group). At this point it is useful to notice the following
properties for gauge propagators

$\displaystyle(\Delta_{i})_{\alpha}^{\ \alpha}$
$\displaystyle=\frac{2(\kappa_{i}D^{2}+p^{2})}{p^{2}(\kappa_{i}^{2}+p^{2})}=\delta_{i},$
(22) $\displaystyle(\Delta_{1})_{\alpha}^{\ \beta}(\Delta_{1})_{\beta}^{\
\gamma}$
$\displaystyle=\frac{2(\kappa_{1}D^{2}+p^{2})}{p^{2}(\kappa_{1}^{2}+p^{2})}(\Delta_{1})_{\alpha}^{\
\gamma}=\delta_{1}(\Delta_{1})_{\alpha}^{\ \gamma},$
$\displaystyle(\Delta_{2})_{\alpha}^{\ \beta}(\Delta_{2})_{\beta}^{\ \gamma}$
$\displaystyle=\frac{2(\kappa_{2}D^{2}+p^{2})}{p^{2}(\kappa_{2}^{2}+p^{2})}(\Delta_{2})_{\alpha}^{\
\gamma}=\delta_{2}(\Delta_{2})_{\alpha}^{\ \gamma},$
$\displaystyle(\Delta_{1})_{\alpha}^{\ \beta}(\Delta_{2})_{\beta}^{\ \gamma}$
$\displaystyle=\delta_{1}(\Delta_{2})_{\alpha}^{\
\gamma}=\delta_{2}(\Delta_{1})_{\alpha}^{\ \gamma}.$

We can then repackage the expression in (21) in terms of the following
matrices:

$\mathcal{M}\,=\,\left(\begin{matrix}M^{AB}&&G^{AN}\\\
(G^{\dagger})^{MB}&&N^{MN}\end{matrix}\right),$ (23)
$\Delta\,=\,\left(\begin{matrix}\delta_{1}\cdot\mathbf{1}_{N_{1}\times
N_{1}}&&0\\\ 0&&\delta_{2}\cdot\mathbf{1}_{N_{2}\times
N_{2}}\end{matrix}\right),$ (24)

where $N_{1}$ and $N_{2}$ are the dimensions of the fundamental
representations of $G_{1}$ and $G_{2}$, respectively. At this point we can
define the generic n-th 1-loop contribution as

$\Sigma^{(n)}_{\text{1-loop}}\equiv\Tr(\mathcal{M}\Delta)^{n}$ (25)

and sum all the contributions to get

$\displaystyle\Sigma$ $\displaystyle=\sum_{n}\mathcal{S}_{n}\
\Sigma^{(n)}_{\text{1-loop}}$ (26)
$\displaystyle=-\frac{1}{2}\Tr(\mathcal{M}\Delta)+\frac{1}{4}\Tr(\mathcal{M}\Delta\mathcal{M}\Delta)-\frac{1}{6}\Tr(\mathcal{M}\Delta\mathcal{M}\Delta\mathcal{M}\Delta)+\,...$
$\displaystyle=-\frac{1}{2}\Tr\log(\mathbb{1}_{(N_{1}+N_{2})\times(N_{1}+N_{2})}+\mathcal{M}\Delta)$
$\displaystyle=-\frac{1}{2}\log\det(\mathbb{1}_{(N_{1}+N_{2})\times(N_{1}+N_{2})}+\mathcal{M}\Delta)$
$\displaystyle=-\frac{1}{2}\log\big(\det\left((\mathbb{1}_{N_{1}\times
N_{1}}+\delta_{1}M)-\delta_{1}\delta_{2}G(\mathbb{1}_{N_{2}\times
N_{2}}+\delta_{2}N)^{-1}G^{T}\right)\cdot\det\left(\mathbb{1}_{N_{2}\times
N_{2}}+\delta_{2}N\right)\big{missing})$

where in the first step we used the fact that
$\mathcal{S}_{n}=\frac{1}{n!}\left(-\frac{1}{2}\right)^{n}(2n-2)!!=\frac{1}{2n}$.

We now switch to the case of $SU(2)\times SU(2)$ gauge group and note few
properties intrinsic to this case, which are helpful to further simplify the
result. In this special case $M$, $N$, and $G$ are all three by three diagonal
matrices therefore they all commute. In fact, by considering the expressions
in (20), we have

$\displaystyle M^{AB}=\frac{g_{1}^{2}\Tr\bar{\Phi}\Phi}{4}\,\delta^{AB}$ (27)
$\displaystyle N^{AB}=\frac{g_{2}^{2}\Tr\bar{\Phi}\Phi}{4}\,\delta^{AB}.$

Hence we obtain

$\Sigma=-\frac{1}{2}\Tr\log\left[\mathbb{1}_{3\times
3}+\delta_{1}M+\delta_{2}N+\delta_{1}\delta_{2}(MN-G^{T}G)\right].$ (28)

Taking into account that the eigenvalues of $G^{T}G$ are given by

$\left(\frac{g_{1}^{2}g_{2}^{2}\det\Phi\det\bar{\Phi}}{4},\frac{g_{1}^{2}g_{2}^{2}\det\Phi\det\bar{\Phi}}{4},\frac{g_{1}^{2}g_{2}^{2}(\Tr\bar{\Phi}\Phi)^{2}}{16}\right)$
(29)

and introducing the notation

$\rho\equiv\Tr\bar{\Phi}\Phi,\qquad B\equiv 2\det\Phi,\qquad\bar{B}\equiv
2\det\bar{\Phi},$ (30)

we can substitute the above results in (28) and obtain

$\Sigma=-\log\left[1+\frac{\rho}{4}(g_{1}^{2}\delta_{1}+g_{2}^{2}\delta_{2})+\frac{\rho^{2}-B\bar{B}}{16}g_{1}^{2}g_{2}^{2}\delta_{1}\delta_{2}\right]-\frac{1}{2}\log\left[1+\frac{\rho}{4}(g_{1}^{2}\delta_{1}+g_{2}^{2}\delta_{2})\right].$
(31)

At this point we can substitute everything in (14) and get the complete 1-loop
expression for the superpotential

$\displaystyle\mathcal{W}_{\text{1-loop}}=$
$\displaystyle-\int\frac{d^{3}p}{(2\pi)^{3}}d^{2}\theta^{\prime}\delta(\theta-\theta^{\prime})\log\left[1+\frac{\rho}{4}(g_{1}^{2}\delta_{1}+g_{2}^{2}\delta_{2})+\frac{\rho^{2}-B\bar{B}}{16}g_{1}^{2}g_{2}^{2}\delta_{1}\delta_{2}\right]\delta^{2}(\theta^{\prime}-\theta)$
(32)
$\displaystyle-\frac{1}{2}\int\frac{d^{3}p}{(2\pi)^{3}}d^{2}\theta^{\prime}\delta(\theta-\theta^{\prime})\log\left[1+\frac{\rho}{4}(g_{1}^{2}\delta_{1}+g_{2}^{2}\delta_{2})\right]\delta^{2}(\theta^{\prime}-\theta).$

Notice now that the expressions between the delta-functions are functions of
$D^{2}$, which enters there via $\delta_{1,2}$. Using the last identity in
(144), one can reduce an arbitrary function of $D^{2}$ to a linear function of
$D^{2}$. Using then the identities

$\displaystyle\delta^{2}(\theta-\theta^{\prime})\delta^{2}(\theta^{\prime}-\theta)=0,$
(33)
$\displaystyle\delta^{2}(\theta-\theta^{\prime})D^{\alpha}\delta^{2}(\theta^{\prime}-\theta)=0,$
$\displaystyle\delta^{2}(\theta-\theta^{\prime})D^{2}\delta^{2}(\theta^{\prime}-\theta)=\delta^{2}(\theta-\theta^{\prime}),$

we rewrite the effective superpotential as

$\displaystyle\mathcal{W}_{\text{1-loop}}=$
$\displaystyle-\int\frac{d^{3}p}{(2\pi)^{3}}\log\left[1+\frac{\rho}{4}(g_{1}^{2}\delta_{1}+g_{2}^{2}\delta_{2})+\frac{\rho^{2}-B\bar{B}}{16}g_{1}^{2}g_{2}^{2}\delta_{1}\delta_{2}\right]\lvert_{D^{2}}$
(34)
$\displaystyle-\frac{1}{2}\int\frac{d^{3}p}{(2\pi)^{3}}\log\left[1+\frac{\rho}{4}(g_{1}^{2}\delta_{1}+g_{2}^{2}\delta_{2})\right]\lvert_{D^{2}}.$

Here $\lvert_{D^{2}}$ means that we should reduce the functions of $D^{2}$ to
linear ones, as explained above, and then take the coefficient in front of
$D^{2}$. In fact, all the coefficients in front of $D^{2}$ terms can be easily
extracted with the aid of the following identity

$(\kappa
D^{2}+p^{2})^{n}\lvert_{D^{2}}=\frac{1}{|p|}\text{Im}\left((i\kappa|p|+p^{2})^{n}\right),$
(35)

which leads us to the final result for the effective superpotential

$\displaystyle\mathcal{W}_{\text{1-loop}}=$
$\displaystyle-\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{|p|}\text{Im}\log\left[1+\frac{\rho}{2}\left(\frac{g_{1}^{2}}{(p^{2}-i\kappa_{1}|p|)}+\frac{g_{2}^{2}}{(p^{2}-i\kappa_{2}|p|)}\right)+\frac{g_{1}^{2}g_{2}^{2}(\rho^{2}-B\bar{B})}{4(p^{2}-i\kappa_{1}|p|)(p^{2}-i\kappa_{2}|p|)}\right]$
(36)
$\displaystyle-\frac{1}{2}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{|p|}\text{Im}\log\left[1+\frac{\rho}{2}\left(\frac{g_{1}^{2}}{(p^{2}-i\kappa_{1}|p|)}+\frac{g_{2}^{2}}{(p^{2}-i\kappa_{2}|p|)}\right)\right].$

In the study of SUSY vacua, the derivatives of $\mathcal{W}_{\text{eff}}$ with
respect to $\rho$, $B$ and $\bar{B}$ are going to be relevant. More precisely,
we will need their asymptotic behavior at large field values. For the general
case of $k_{1}\neq-k_{2}$ they are computed to be

$\displaystyle\partial_{\rho}\mathcal{W}_{\text{1-loop}}$
$\displaystyle=\begin{cases}-\frac{F_{1}+F_{4}}{\rho^{1/2}}\,+\,\mathcal{O}\left(\rho^{-3/2},(\frac{B}{\rho})^{2}\right)\quad\text{if}\
\frac{B}{\rho}\ll 1,\\\
-G-\frac{3F_{1}+F_{2}}{\rho^{1/2}}\,+\,\mathcal{O}\left(\rho^{-3/2}\right)\
\quad\text{if}\ B=\rho\end{cases}$ (37)
$\displaystyle\partial_{B}\mathcal{W}_{\text{1-loop}}$
$\displaystyle=\begin{cases}\frac{F_{3}B}{\rho^{3/2}}\,+\,B\,\mathcal{O}\left(\rho^{-5/2},\frac{B}{\rho}\right)\
\quad\text{if}\ \frac{B}{\rho}\ll 1,\\\
G+\frac{F_{2}}{\rho^{1/2}},+\,\mathcal{O}\left(\rho^{-3/2}\right)\qquad\text{if}\
B=\rho.\end{cases}$

Here $F_{1},F_{2},F_{3},F_{4},G_{1}$ are functions of
$g_{1},g_{2},\kappa_{1},\kappa_{2}$:

$\displaystyle F_{1}$
$\displaystyle=\frac{\kappa_{1}g_{1}^{2}+\kappa_{2}g_{2}^{2}}{16\sqrt{2}\pi\sqrt{g_{1}^{2}+g_{2}^{2}}},$
(38a) $\displaystyle F_{2}$
$\displaystyle=\frac{g_{1}^{2}g_{2}^{2}\left((\kappa_{1}+\kappa_{2})(g_{1}^{2}+g_{2}^{2})-3(\kappa_{1}g_{2}^{2}+\kappa_{2}g_{1}^{2})\right)}{4\sqrt{2}\pi(g_{1}^{2}+g_{2}^{2})^{5/2}},$
(38b) $\displaystyle F_{3}$
$\displaystyle=\frac{g_{1}g_{2}(\kappa_{1}g_{2}+\kappa_{2}g_{1})}{4\sqrt{2}\pi(g_{1}+g_{2})^{2}},$
(38c) $\displaystyle F_{4}$
$\displaystyle=\frac{g_{1}\kappa_{1}+g_{2}\kappa_{2}}{8\sqrt{2}\pi}$ (38d)
$\displaystyle G$
$\displaystyle=\frac{g_{1}^{2}g_{2}^{2}}{4\pi(g_{1}^{2}+g_{2}^{2})}.$ (38e)

For the case of $k_{1}=-k_{2}$ one can go a bit further and get not only the
leading asymptotics, but complete closed form expressions for
$\partial_{\rho}\mathcal{W}_{\text{1-loop}}$ and
$\partial_{|B|}\mathcal{W}_{\text{1-loop}}$ in the limits $B\rightarrow\rho$
and $B\rightarrow 0$: these results are collected in Appendix C.

As is explained in Bashmakov:2018wts , the asymptotic behaviour of the
effective superpotential is fully determined by the one-loop contributions and
does not receive higher-order corrections. This fact implies that the results
in (37),(38) are actually exact.

##  4 Phase diagrams of $SU(2)\times SU(2)$ models

We are now ready to apply the results of the previous section to the study of
the IR phases of the $SU(2)\times SU(2)$ quiver theories. We will add the
tree-level mass term to the superpotential, such that the total superpotential
is given by the sum of the mass term and the one-loop effective
superpotential,

$\mathcal{W}=m\Tr\bar{\Phi}\Phi+\mathcal{W}_{1-loop}.$ (39)

With the effective superpotential at hands, we can proceed and study
supersymmetric vacua of the theory, which are given by the critical points of
the superpotential,

$\bar{\partial}\mathcal{W}\,=\,0.$ (40)

As was discussed above (4), we can apply two gauge transformations and put the
scalar matrix into the form

$\Phi=\left(\begin{matrix}\phi_{11}&&0\\\ 0&&\phi_{22}\end{matrix}\right),$
(41)

where the common phase of $\phi_{11}$ and $\phi_{22}$ have been set to zero
thanks to the $U(1)$ baryonic symmetry. The equation in (40) can be expanded
as

$\displaystyle\phi_{11}\partial_{\rho}\mathcal{W}\,+\,\phi_{22}\partial_{|B|}\mathcal{W}\,=\,0,$
(42)
$\displaystyle\phi_{22}\partial_{\rho}\mathcal{W}\,+\,\phi_{11}\partial_{|B|}\mathcal{W}\,=\,0.$

From the equations above one can then easily infer the following
possibilities:

  * 1)

The vacuum at the origin, $\phi_{11}=\phi_{22}=0$.

  * 2)

$\partial_{\rho}\mathcal{W}=-\partial_{|B|}\mathcal{W}\neq 0$,
$\phi_{11}=\phi_{22}$. 141414There is also a possibility that
$\partial_{\rho}\mathcal{W}=\partial_{|B|}\mathcal{W}\neq 0$,
$\phi_{11}=-\phi_{22}$, but this is gauge-equivalent to the one stated above.

  * 3)

$\partial_{\rho}\mathcal{W}=\partial_{|B|}\mathcal{W}=0$.

At this point it is convenient to split the discussion into two different
cases: Chern-Simons level of the same sign and Chern-Simons level of opposite
signs.

###  4.1 Chern-Simons levels of the same sign

In this subsection we deal with $\mathcal{N}=1$ $SU(2)_{k_{1}}\times
SU(2)_{k_{2}}$ coupled to a bi-findamental multiplet, and assume that $0\leq
k_{1}\leq k_{2}$151515The situation of $0\leq k_{2}\leq k_{1}$ is obtained by
exchanging two nodes and the situation of $0\geq k_{2}\geq k_{1}$ is obtained
by applying the time reversal transformation.. We postpone the study of the
$k_{1}=k_{2}=0$ case, which requires a separate treatment. When the mass
parameter is large and positive, we can integrate the matter out and obtain a
single vacuum with the infrared theory given by

$\mathcal{N}=1\qquad SU(2)_{k_{1}+1}\times SU(2)_{k_{2}+1}.$ (43)

This theory preserves supersymmetry and further flows to a topological CS
theory in the IR (see Section 2.3),

$SU(2)_{k_{1}}\times SU(2)_{k_{2}}\quad\text{TQFT}.$ (44)

The Witten index is (taking into account that there are six negative-mass
Majorana gaugini)

$\text{WI}_{+}=(k_{1}+1)(k_{2}+1).$ (45)

If the mass parameter is large and negative, we can again integrate the matter
out, but with the result

$\mathcal{N}=1\qquad SU(2)_{k_{1}-1}\times SU(2)_{k_{2}-1}.$ (46)

The fate of this vacuum now depends on the values of $k_{1}$ and $k_{2}$.

  * •

If $k_{1},k_{2}>1$, the vacuum preserves supersymmetry and flows to a CS
theory.

$SU(2)_{k_{1}-2}\times SU(2)_{k_{2}-2}\quad\text{TQFT},$ (47)
$\text{WI}_{-}=(k_{1}-1)(k_{2}-1).$ (48)

  * •

If $k_{1}=1$, $k_{2}>1$, the vacuum breaks supersymmetry, and we get a
Majorana goldstino together with a decoupled CS theory,

$G_{\alpha}\,+\,U(1)_{2}\times SU(2)_{k_{2}-2}\quad\text{TQFT}.$ (49)

  * •

If $k_{1}=k_{2}=1$, the vacuum breaks supersymmetry, and the IR theory is

$G_{\alpha}\,+\,U(1)_{2}\times U(1)_{2}\quad\text{TQFT}.$ (50)

  * •

If $k_{1}=0$, $k_{2}>1$, supersymmetry is preserved, and the IR theory is

$SU(2)_{k_{2}-2}\quad\text{TQFT},$ (51) $\text{WI}_{-}=-(k_{2}-1).$ (52)

  * •

If $k_{1}=0$, $k_{2}=1$, supersymmetry is again broken, and we get in the IR

$G_{\alpha}\,+\,U(1)_{2}\quad\text{TQFT}.$ (53)

We note in particular that the large positive mass phase and the large
negative mass phase have different Witten indices. In order to understand the
transition between the large negative mass phase and the large positive mass
phase, it is useful to understand the dynamics near the point $m=0$, where the
asymptotic behaviour of the superpotential changes and the Witten index can
jump.

We start by observing that the vacuum of the first kind, namely at the origin
of the field space, $\phi_{11}=\phi_{22}=v_{1}=0$, exists for $m=0$ as well as
for $m$ small and positive or small and negative. We identify this vacuum with
the semiclassical vacuum we have seen at large and negative mass. (It will be
evident in a moment that this choice leads to a consistent phase diagram
providing the matching of the Witten index at the phase transition locus).
This vacuum either preserves supersymmetry or, if one of the CS levels is
equal to one, breaks it non-perturbatively. It is then expected to find new
vacua appearing from the infinity of the field space near the line $m=0$, and
whose total Witten index must be different from zero. We thus initiate the
search of these vacua, which must be either of the second or of the third
type.

####  4.1.1 Non-Abelian vacuum

We first turn to the analysis of the vacuum of the second kind, with
$\phi_{11}=\phi_{22}=v_{2}$ for some real and positive $v_{2}$, and with
$\rho=B=\bar{B}=2v_{2}^{2}$. Then the equation
$\partial_{\rho}\mathcal{W}=-\partial_{|B|}\mathcal{W}$ turns into

$-G-\frac{3F_{1}+F_{2}}{\rho^{1/2}}+m=-G-\frac{F_{2}}{\rho^{1/2}}$ (54)

We immediately notice that, thanks to the positivity of $F_{1}$ (see (38)), we
must necessarily have $m>0$. Therefore, when we move from the negative-mass
region and cross the $m=0$ line, a new vacuum is found, in precise accordance
with our expectations. The expression for the vev can be then easily found and
is given by

$v_{2}=\frac{3F_{1}}{\sqrt{2}m}$ (55)

Next we determine the effective low-energy theory of this vacuum. The vev

$\Phi=\left(\begin{matrix}v_{2}&&0\\\ 0&&v_{2}\end{matrix}\right)$ (56)

breaks the global baryonic symmetry $U(1)_{B}$, thus we expect to see the
corresponding Goldstone boson with its superpartner. The vacuum also breaks
the gauge group to the diagonal $SU(2)$ with the induced CS level equal to
$k_{1}+k_{2}$. The CS level can receive quantum corrections when massive
fermions charged under the unbroken $SU(2)$ are integrated out, so we need to
understand the fermionic mass spectrum.

The fermionic mass terms originate from the superpotential, from the gaugini-
matter coupling terms, and from the gaugini mass term, a supersymmetric
counterpart of the CS term:

$\displaystyle\mathcal{L}_{\psi^{2}}=$
$\displaystyle\frac{\partial^{2}\mathcal{W}}{\partial{\bar{\Phi}_{\hat{i}j}\partial\Phi_{k\hat{l}}}}\bar{\Psi}_{\hat{i}j}\Psi_{k\hat{l}}\,+\frac{1}{2}\left(\,\frac{\partial^{2}\mathcal{W}}{\partial{\bar{\Phi}_{\hat{i}j}\partial\bar{\Phi}_{\hat{k}l}}}\bar{\Psi}_{\hat{i}j}\Psi^{c}_{\hat{k}l}\,+\,\text{c.c.}\right)\,+\,\left(ig_{1}\text{Tr}\bar{\Psi}\lambda_{1}\Phi\,+\,ig_{2}\text{Tr}\bar{\Psi}^{c}\lambda_{2}\bar{\Phi}\,+\,\text{c.c.}\right)+$
(57)
$\displaystyle-\kappa_{1}\text{Tr}\bar{\lambda}_{1}\lambda_{1}\,-\,\kappa_{2}\text{Tr}\bar{\lambda}_{2}\lambda_{2},$

where the indices are put on the same line for convenience, and scalars are
assumed to take their vev.

While we are given fermions in representations of $SU(2)\times SU(2)$ group,
it is natural to decompose them into representations of the diagonal $SU(2)$
group. The decomposition goes as follows,

$\Psi_{i}^{\ \hat{j}}=\frac{1}{2}\left[\psi^{a}_{\text{Re}}(\sigma^{a})_{i}^{\
\hat{j}}+i\psi^{a}_{\text{Im}}(\sigma^{a})_{i}^{\
\hat{j}}+(\psi_{0}+i\psi_{G})\delta_{i}^{\ \hat{j}}\right],$ (58)

where we introduced two Majorana modes, $\psi_{0}$ and $\psi_{G}$, neutral
under the diagonal $SU(2)$, and two Majorana multiplets,
$\psi_{\text{Re}}^{a}$ and $\psi_{\text{Im}}^{a}$, transforming in the adjoint
representation of $SU(2)$. Two other adjoint multiplets are provided by
$\lambda_{1}$ and $\lambda_{2}$.
It follows from (57) that the mass of $\psi_{0}$ and $\psi_{G}$ are

$\displaystyle m_{0}$
$\displaystyle=4v_{2}^{2}\left(\partial^{2}_{\rho}\mathcal{W}+2\partial_{\rho}\partial_{B}\mathcal{W}+\partial^{2}_{B}\mathcal{W}\right)=m,$
(59a) $\displaystyle m_{G}$ $\displaystyle=0,$ (59b)

thus we can identify the latter with the superpartner of the Goldstone boson
associated with the broken $U(1)_{B}$. The mass of $\psi_{\text{Re}}$ instead
reads

$m_{\text{Re}}=2\partial_{\rho}\mathcal{W}=-2G-2m\left(1+\frac{F_{2}}{3F_{1}}\right).$
(60)

Notice that, since we must have that $m>0$ from (54) and $G>0$ by definition,
it follows that $m_{\text{Re}}$ must be negative.
Finally, $\psi_{\text{Im}}$, $\lambda_{1}$, and $\lambda_{2}$ mix with each
other via the mass matrix

$\left(\begin{matrix}0&&g_{1}v_{2}&&-g_{2}v_{2}\\\
g_{1}v_{2}&&-\kappa_{1}&&0\\\ -g_{2}v_{2}&&0&&-\kappa_{2}\end{matrix}\right).$
(61)

This mass matrix has one positive eigenvalue and two negative eigenvalues (one
of the eigenmodes with a negative eigenvalue can be identified with the
gaugino of the unbroken gauge group). Therefore, in total we have three
adjoint multiplets with negative mass and one adjoint multiplet with positive
mass, coupled to the unbroken $SU(2)$: integrating them out, we get a shift of
the CS level equal to $-2$.

We are thus ready to formulate the infrared dynamics of this vacuum. Unless
$k_{1}=0$ and $k_{2}=1$, it preserves supersymmetry, and the infrared dynamics
is described by

$\Phi_{G}\,+\,SU(2)_{k_{1}+k_{2}-2}\quad\text{TQFT},$ (62)

where $\Phi_{G}$ is the Goldstone supermultiplet. If, on the contrary,
$k_{1}=0$ and $k_{2}=1$, supersymmetry is spontaneously broken, and we get in
the infrared

$\phi_{G}\,+\,G_{\alpha}\,+\,U(1)_{2}\quad\text{TQFT},$ (63)

with $\phi_{G}$ being the Goldstone boson.

####  4.1.2 Abelian vacuum

Next we consider the vacuum of the third kind appearing from infinity in the
field space. The condition $\partial_{B}\mathcal{W}=0$ requires that
$B=0$161616One can first observe that $B\leq\rho$, and from (37) we see that
for $B=\rho\rightarrow\infty$, $\partial_{B}\mathcal{W}>0$, while for $B=0$,
$\partial_{B}\mathcal{W}=0$, as desired. It can then be examined numerically
that there are no other solutions for $0<B<\rho$.. The condition
$\partial_{\rho}\mathcal{W}$ then reads as

$m-\frac{F_{1}+F_{4}}{\rho^{1/2}}=0.$ (64)

Since $F_{1}+F_{4}>0$, there are no solutions for $m\leq 0$, while for $m>0$
we find

$\Phi=\left(\begin{matrix}v_{3}&&0\\\ 0&&0\end{matrix}\right),\qquad
v_{3}=\frac{F_{1}+F_{4}}{m}.$ (65)

The next step is to study the IR dynamics of this vacuum. The $U(1)_{B}$ is
preserved here, and is generated by 171717We note that this choice of the
preserved $U(1)_{B}$ is not unique, and is defined up to an action of the
gauge transformation.

$\frac{i}{2}\left(\begin{matrix}1&&0\\\
0&&1\end{matrix}\right)\oplus\frac{i}{2}\left(\begin{matrix}1&&0\\\
0&&-1\end{matrix}\right)_{2},$ (66)

and the gauge symmetry is broken to $U(1)$, generated by

$\frac{i}{2}\left(\begin{matrix}1&&0\\\
0&&-1\end{matrix}\right)_{1}\oplus\frac{i}{2}\left(\begin{matrix}1&&0\\\
0&&-1\end{matrix}\right)_{2}.$ (67)

where the subscripts $1,2$ indicate respectively the first or the second gauge
group factor. This Abelian gauge field inherits the Chern-Simons level
$2(k_{1}+k_{2})$.
We can now classify fermionic modes according to their charges with respect to
the unbroken $U(1)\times U(1)_{B}$. The matter multiplet fermions and two
types of gaugini can be decomposed as

$\Psi=\left(\begin{matrix}\frac{\eta+i\chi}{\sqrt{2}}&&\psi_{+}\\\
\psi_{-}&&\psi_{0}\end{matrix}\right),\qquad\lambda_{1}=\frac{1}{2}\left(\begin{matrix}\lambda_{1,0}&&\sqrt{2}\lambda_{1,+}\\\
\sqrt{2}\lambda^{c}_{1,+}&&-\lambda_{1,0}\end{matrix}\right),\qquad\lambda_{2}=\frac{1}{2}\left(\begin{matrix}\lambda_{2,0}&&\sqrt{2}\lambda_{2,+}\\\
\sqrt{2}\lambda^{c}_{2,+}&&-\lambda_{2,0}\end{matrix}\right).$ (68)

All the types of the modes (Majorana or Dirac) as well as their charges are
summarized in Table 1.

Mode | $\eta$ | $\chi$ | $\psi_{0}$ | $\psi_{+}$ | $\psi_{-}$ | $\lambda_{1,0}$ | $\lambda_{1,+}$ | $\lambda_{2,0}$ | $\lambda_{2,+}$
---|---|---|---|---|---|---|---|---|---
Type | M | M | D | D | D | M | D | M | D
$U(1)$ | 0 | 0 | 0 | 1 | -1 | 0 | 1 | 0 | 1
$U(1)_{B}$ | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1
Table 1: Fermion modes

The masses of fermions neutral under the $U(1)$ gauge group are determined as
follows. By starting again from (57), we get that $\eta$ and $\psi_{0}$ masses
come from the superpotential,

$\displaystyle
m_{\eta}=2v_{3}^{2}\frac{\partial^{2}\mathcal{W}}{\partial\rho^{2}}=m,$ (69)
$\displaystyle
m_{\psi_{0}}=v_{3}^{2}\left(\frac{\partial^{2}\mathcal{W}}{\partial
B^{2}}+\frac{1}{B}\frac{\partial\mathcal{W}}{\partial
B}\right)=2v_{3}^{2}\frac{\partial^{2}\mathcal{W}}{\partial
B^{2}}=2m\frac{F_{3}}{F_{1}+F_{4}},$

whereas $\chi$, $\lambda_{1,0}$, $\lambda_{2,0}$ mix with each other through
the mass matrix

$\left(\begin{matrix}0&&\frac{1}{\sqrt{2}}g_{1}v_{3}&&-\frac{1}{\sqrt{2}}g_{2}v_{3}\\\
\frac{1}{\sqrt{2}}g_{1}v_{3}&&-\kappa_{1}&&0\\\
-\frac{1}{\sqrt{2}}g_{2}v_{3}&&0&&-\kappa_{2}\end{matrix}\right).$ (70)

There are also mixing modes charged under the $U(1)$ gauge group, in
particular, $\psi_{-}$ mixes with $\lambda^{c}_{1,+}$ via

$\left(\begin{matrix}0&&i\frac{1}{\sqrt{2}}g_{1}v_{3}\\\
-i\frac{1}{\sqrt{2}}g_{1}v_{3}&&-\kappa_{1}\end{matrix}\right),$ (71)

and $\psi_{+}$ mixes with $\lambda_{2,+}$ via

$\left(\begin{matrix}0&&-i\frac{1}{\sqrt{2}}g_{2}v_{3}\\\
i\frac{1}{\sqrt{2}}g_{2}v_{3}&&-\kappa_{2}\end{matrix}\right).$ (72)

Both matrices have one positive eigenvalue and one negative eigenvalue which
implies that the $U(1)$ CS level does not get renormalized when these massive
modes are integrated out. We thus conclude that at low energies we get a pure
Abelian CS theory

$U(1)_{2(k_{1}+k_{2})}\quad\text{TQFT}.$ (73)

${\ast}\,{\ast}\,{\ast}$

Few comments are in order. First, we were able to follow the appearance of two
new vacua as far as the line $m=0$ is crossed. This process is controlled just
by the leading asymptotic of the effective superpotential, which in turn is
determined only by the one-loop contribution. We thus conclude that we have
rigorously derived the existence of these vacua. Second, there might be
supersymmetric vacua emerging for some values of the parameters
$g_{1},g_{2},m$, which don’t come from infinity, but rather appear at finite
field values. These vacua should have vanishing total Witten index, and their
dynamics is a priori governed by perturbation theory at all orders, and not
just at one-loop level. We do not have reliable tools to study them and,
moreover, there are no consistency requirements (i.e. Witten index matching)
that would necessitate their existence.

####  4.1.3 Summary of the results and Phase Diagrams

We are now able to formulate the phase diagram of the theory under
consideration. We start with a generic case of $k_{1}>1$ and $k_{2}>1$, and
the relevant phase diagram is schematically depicted in Figure 2, where we
attempt to reflect only the topology of the phase diagram181818In particular,
various straight lines appearing on the figure should in practice be curved..
At large and negative masses, and up to the $m=0$ line there is the
supersymmetric semiclassical vacuum described by

$SU(2)_{k_{1}-2}\times SU(2)_{k_{2}-2}\quad\text{TQFT},$ (74)

as we saw in (47). This vacuum is denoted on the figure as $v_{1}^{-}$, and
this phase corresponds to the purple region. The Witten index of this vacuum
is

$\text{WI}_{1}=(k_{1}-1)(k_{2}-1).$ (75)

When we cross the wall at $m=0$, two new vacua come in from infinity,

$\displaystyle\Phi_{G}\,+\,SU(2)_{k_{1}+k_{2}-2}\quad\text{TQFT},\quad\text{WI}_{2}=0,$
(76) $\displaystyle
U(1)_{2(k_{1}+k_{2})}\quad\text{TQFT},\quad\text{WI}_{3}=2(k_{1}+k_{2}),$

giving us a phase with three vacua (the new vacua are $v_{2}$ and $v_{3}$, and
the corresponding region in Figure 2 is the light blue one). These three vacua
must undergo, generically, two second-order phase transitions (lines $m_{*}$
and $m_{**}$, which are actually functions of $g_{1},g_{2}$), merging into a
single vacuum seen at large positive masses which we indicate with
$v_{1}^{+}$.

These phase transitions are supposed to happen somewhere around the origin of
the field space, where the physics is strongly coupled, and we do not have
much control over it. The structure of vevs in each vacuum suggests (see the
figure on the right) that at the first phase transition either $v_{2}$ merges
with $v_{1}^{-}$, or $v_{3}$ merges with $v_{1}^{-}$: we conjecture, basing on
a duality proposed in Section 6, that the first option is realized. In the
intermediate phase we then still get the Abelian vacuum $v_{3}$ and some other
vacuum, $v_{q}$, which is guessed to support again the $SU(2)_{k_{1}-2}\times
SU(2)_{k_{2}-2}$ CS theory (the yellow region in Figure 2). At the second
phase transition two vacua merge and produce the large positive mass vacuum
$v_{1}^{+}$.

Figure 2: Structure of the phase diagrams of the $SU(2)_{k_{1}}\times
SU(2)_{k_{2}}$ quivers and, as will be clear later, the $SU(2)_{k_{1}}\times
U(2)_{k_{2},k_{3}}$ quivers with either $k_{1},k_{2}>1$, or $k_{1}=0$,
$k_{2}>1$. Dashed lines correspond to the second order phase transitions,
while the solid line is the wall. Supersymmetric vacua at each phase are
indicated.

Let us now discuss some exceptional cases with low values of the CS levels.

  * •

If $k_{1}=1$, $k_{2}\geq 1$, the negative mass vacuum breaks supersymmetry. As
soon as the wall $m=0$ is crossed, two supersymmetric vacua appear,

$\displaystyle\Phi_{G}\,+\,SU(2)_{k_{2}-1}\quad\text{TQFT},\quad\text{WI}_{2}=0,$
(77) $\displaystyle
U(1)_{2(k_{2}+1)}\quad\text{TQFT},\quad\text{WI}_{3}=2(k_{2}+1),$

at some value of the mass parameter they merge and give rise to the large mass
vacuum,

$SU(2)_{1}\times SU(2)_{k_{2}}\quad\text{TQFT}.$ (78)

The picture is illustrated in Figure 3.

Figure 3: Phase diagram for $k_{1}=1$, $k_{2}\geq 1$. Dashed line correspond
to the second order phase transition, while the solid line is the wall.
Supersymmetric vacua at each phase are indicated.

  * •

If $k_{1}=0$ and $k_{2}>1$, the negative mass vacuum is supersymmetric and the
theory flows to

$SU(2)_{k_{2}-2}\quad\text{TQFT},\quad\text{WI}_{0}=-(k_{2}-1).$ (79)

When we cross the line $m=0$, two supersymmetric vacua appear:

$\displaystyle\Phi_{G}\,+\,SU(2)_{k_{2}-2}\quad\text{TQFT},\quad\text{WI}_{2}=0,$
(80) $\displaystyle U(1)_{2k_{2}}\quad\text{TQFT},\quad\text{WI}_{3}=2k_{2},$

When $m$ is increased, the resulting three vacua undergo two phase transition
and produce a supersymmetric vacuum with

$SU(2)_{k_{2}}\quad\text{TQFT}.$ (81)

Figure 4: Phase diagram for $k_{1}=0$, $k_{2}=1$. The solid line correspond to
the wall. The supersymmetric vacum is indicated.

  * •

If $k_{1}=0$ and $k_{2}=1$, the negative mass vacuum is not supersymmetric.
When the wall is crossed, there appears one SUSY-breaking vacuum (see (63))
and one supersymmetric vacuum with

$U(1)_{2}\quad\text{TQFT}$ (82)

in the IR. Using the chain of dualities for TQFTs, we observe

$U(1)_{2}\ \longleftrightarrow\ U(1)_{-2}\longleftrightarrow\ SU(2)_{1},$ (83)

and so the new supersymmetric vacuum came in from infinity exactly reproduces
the semiclassical large mass vacuum. Therefore, in this case the theory does
not undergo any phase transitions. (See Figure 4)

An important check of the picture that we are suggesting here is the matching
of the Witten indices upon the phase transition points. As an example, in the
case of $k_{1},k_{2}>1$ we have

$\text{WI}_{1}+\text{WI}_{2}+\text{WI}_{3}=(k_{1}-1)(k_{2}-1)+0+2(k_{1}+k_{2})=(k_{1}+1)(k_{2}+1).$
(84)

The right-hand side is nothing else but the Witten index of the large mass
vacuum.

###  4.2 Chern-Simons levels of opposite signs

Having understood the phase diagram for the case when two CS levels have the
same sign, we now turn to the situation when the first CS level is positive
and the second is negative. We thus consider the theory

$SU(2)_{k_{1}}\times SU(2)_{-k_{2}},$ (85)

coupled as before to a bi-fundamental matter $\Phi$, with $k_{1},k_{2}>0$, and
we will also assume w.l.g. that $k_{1}>k_{2}$. The results of the following
discussion are summarized in Figure 5.

As before, it is useful to start the analysis with considering the large mass
semiclassical phases. For large and negative masses we find a supersymmetric
vacuum with

$SU(2)_{k_{1}-2}\times SU(2)_{-k_{2}}\quad\text{TQFT}$ (86)

in the IR ($v_{1}^{-}$ in Figure 5). In the large positive mass phase we see
the following picture:

  * •

When $k_{2}>1$, we get a supersymmetric vacuum ($v_{1}^{+}$ on Figure (5))
hosting a CS theory,

$SU(2)_{k_{1}}\times SU(2)_{-k_{2}+2}\quad\text{TQFT}.$ (87)

  * •

When $k_{2}=1$, SUSY gets broken, and the IR theory is given by

$G_{\alpha}+SU(2)_{k_{1}}\times U(1)_{2}\quad\text{TQFT}.$ (88)

We first discuss in details the case of $k_{2}>1$, and then comment on the
changes in the picture when $k_{2}=1$.

As soon as the large mass phases are understood, the next step is to study the
behaviour near the wall, at $m=0$. Again, we see that the
$\phi_{11}=\phi_{22}=0$ vacuum exists on both sides of the wall. It will again
be natural to identify this vacuum with one of the large mass vacua, however
this time it is less obvious to decide which of the two should be chosen. We
also remark that, while moving along the line $m=0$, three special points can
be distinguished. These are the points for which the asymptotic behaviour of
the effective superpotential (37) changes in a certain way, and are given by

$\displaystyle
F_{1}=0,\quad\frac{g_{1}}{g_{2}}=\left(\frac{k_{2}}{k_{1}}\right)^{\frac{1}{4}},$
(89a) $\displaystyle F_{1}+F_{4}=0,\quad\frac{g_{1}}{g_{2}}=\alpha,$ (89b)
$\displaystyle F_{3}=0,\quad\frac{g_{1}}{g_{2}}=\frac{k_{2}}{k_{1}}.$ (89c)

Here $\alpha$ is the single positive root of the equation
$x^{4}-\frac{k_{2}}{k_{1}}+2(x^{3}-\frac{k_{2}}{k_{1}})\sqrt{x^{2}+1}=0$, and
we note that for $k_{1}>k_{2}$ we have

$\frac{k_{2}}{k_{1}}<\alpha<\left(\frac{k_{2}}{k_{1}}\right)^{\tfrac{1}{4}}$
(90)

As we did in the previous subsection, we now give a detailed discussion of the
vacuuum structure near the wall.

####  4.2.1 Non-Abelian vacuum

We first search for the vacuum of the second type. The equation
$\rho=B=\bar{B}=2v_{2}^{2}$ again reduces to

$-\frac{3F_{1}}{\rho^{1/2}}+m=0,$ (91)

but now $F_{1}$ changes the sign when $g_{1},g_{2}$ are varied, indeed what
happens is that

$F_{1}\gtrless
0,\quad\frac{g_{1}}{g_{2}}\gtrless\left(\frac{k_{2}}{k_{2}}\right)^{1/4}.$
(92)

We thus conclude that a vacuum of the second type is still given by

$v_{2}=\frac{3F_{1}}{\sqrt{2}m},$ (93)

and exists when either
$m>0,\,\tfrac{g_{1}}{g_{2}}>\left(\tfrac{k_{2}}{k_{2}}\right)^{1/4}$ or
$m<0,\,\tfrac{g_{1}}{g_{2}}<\left(\tfrac{k_{2}}{k_{2}}\right)^{1/4}$ ($v_{2}$
in the orange, light blue, grey and brown regions of Figure 5). At the point
$\tfrac{g_{1}}{g_{2}}=\left(\tfrac{k_{2}}{k_{2}}\right)^{1/4}$ the quantum
potential develops an asymptotic direction with zero energy. This is the first
special point mentioned above.

The gauge and global symmetry breaking pattern in this vacuum is the same as
for the CS levels of the same sign, $U(1)_{B}$ is spontaneously broken, and
the unbroken gauge group is $SU(2)_{k_{1}-k_{2}}$. We can also apply the
previously obtained results for the fermionic mass spectrum, which does not
undergo any changes. The resulting low-energy theory again depends on the
values of the levels.

  * •

When $k_{1}>k_{2}+1$, supersymmetry is preserved, and at low energies we get
the Goldstone multiplet and a CS theory,

$\Phi_{G}\,+\,SU(2)_{k_{1}-k_{2}-2}\quad\text{TQFT}.$ (94)

  * •

When $k_{1}=k_{2}+1$, supersymmetry is broken, and we get in the IR

$\phi_{G}\,+\,G_{\alpha}\,+\,U(1)_{2}\quad\text{TQFT},$ (95)

where $\phi_{G}$ is the Goldstone boson.

####  4.2.2 Abelian vacua

Finally, we look for Abelian vacua. There is still a solution given by

$\Phi=\left(\begin{matrix}v_{3}&&0\\\ 0&&0\end{matrix}\right),\quad
v_{3}=\frac{F_{1}+F_{4}}{m}.$ (96)

Introducing the critical value $\frac{g_{1}}{g_{2}}=\alpha$ such that
$F_{1}(\alpha)+F_{4}(\alpha)=0$, we see that the solution exists either for
$m>0,\,\tfrac{g_{1}}{g_{2}}>\alpha$, or for
$m<0,\,\tfrac{g_{1}}{g_{2}}<\alpha$. At the point
$\tfrac{g_{1}}{g_{2}}=\alpha$ the quantum superpotential again developes an
asymptotic direction with zero energy.

Figure 5: Structure of the phase diagrams of the $SU(2)_{k_{1}}\times
SU(2)_{-k_{2}}$ quivers and, as will be clear later, the $SU(2)_{k_{1}}\times
U(2)_{-k_{2},-k_{3}}$ quivers with $k_{1},k_{2}>0$ and $k_{1}>k_{2}$. Dashed
lines correspond to the second order phase transitions, while the solid line
is the wall. Supersymmetric vacua at each phase are indicated.

We also note that, as follows from the definition of $F_{4}$,
$\alpha<\left(\tfrac{k_{2}}{k_{1}}\right)^{1/4}$. This means that when we
gradually move along the $m=0$ line from the region with $g_{1}\gg g_{2}$ to
the region with $g_{1}\ll g_{2}$, we first see the flipping point for the Non-
Abelian vacuum (where it moves from the positive mass region to the negative
mass region), and then the flipping point for the Abelian vacuum.

To determine the IR physics of this vacuum, one has to reexamine the fermionic
mass spectrum. It follows from (69)-(72) that upon passing by the point
$\frac{g_{1}}{g_{2}}=\alpha$, the charged modes do not change mass signs (but
one neutral Dirac mode does), thus the IR description is given by

$U(1)_{2(k_{1}-k_{2})}\quad\text{TQFT}.$ (97)

Yet, this is not the end of the story. Indeed, when the point
$\frac{g_{1}}{g_{2}}=\frac{k_{2}}{k_{1}}$ is passed by, a new solution for the
equations $\partial_{|B|}\mathcal{W}=\partial_{\rho}\mathcal{W}=0$ is found.
This can be seen in the following way. We note that $0\leq|B|\leq\rho$, and

$\displaystyle\partial_{|B|}\mathcal{W}\,\vline_{|B|=\rho}>0,$
$\displaystyle\partial_{|B|}\mathcal{W}\,\vline_{|B|=0}=0$

for any values of the parameters. But $\partial^{2}_{|B|}\mathcal{W}\propto
F_{3}$ changes the sign exactly at the point
$\frac{g_{1}}{g_{2}}=\frac{k_{2}}{k_{1}}$. In fact, when
$\frac{g_{1}}{g_{2}}>\frac{k_{2}}{k_{1}}$,
$\partial^{2}_{|B|}\mathcal{W}\,\vline_{|B|=0}>0$, and so it is possible that
$B=0$ is the only zero of $\partial_{|B|}\mathcal{W}=0$: indeed, this is
confirmed by the numerical study of the superpotential. On the other hand,
when $\frac{g_{1}}{g_{2}}<\frac{k_{2}}{k_{1}}$,
$\partial^{2}_{|B|}\mathcal{W}\,\vline_{|B|=0}<0$, and so there is at least
one more solution with $B\neq 0$: the numerical study confirms that there is
indeed only one such solution. This new vacuum still breaks the gauge group
down to $U(1)$, but it also breaks the global $U(1)_{B}$. The IR phyisc is
thus represented by

$\Phi_{G}+U(1)_{2(k_{1}-k_{2})}\quad\text{TQFT}.$ (99)

####  4.2.3 Phase Diagrams

We now summarize the picture we suggest for the phase diagram, starting from
the case $k_{1},k_{2}>1$ and keeping in mind the relation in (90). We recall
that we started by determining the large mass phases, depicted by the purple
and the green regions of Figure 5. The next step was to understand the near-
the-wall behaviour. When $g_{1}\gg g_{2}$, there is just one vacuum on the
left from the wall, but two new vacua, the Non-Abeian and the Abelian ones,
appear on the right from the wall (orange regions of Figure 5). While moving
down along the wall, we encounter the first special point, after which the
Non-Abelian vacuum is found on the left from the wall, while the Abelian
vacuum is still on the right: this corresponds to the light blue and the
yellow regions. In both these phases there is also a vacuum at the origin. We
do not have a weak coupling limit that would allow the direct study of this
vacuum, but we propose that its IR description is identical to the large
negative mass vacuum, $v_{1}^{-}$, since it provides the correct Witten index,
and automatically matches the UV 1-form symmetry ’t Hooft anomaly.

When we decreese $\frac{g_{1}}{g_{2}}$ even further, the second special point
is found. While passing it, we find that there is just one vacuum on the right
from the wall (the green phase in Figure 5), and three vacua on the left: a
vacuum at the origin together with the Non-Abelian and the Abelian vacua
discussed above (the grey phase in Figure 5). The vacuum at the origin is now
identified with the large mass vacuum.

If we go even further down the wall, the Abelian vacuum splits into two
Abelain vacua ($v_{3}^{-}$ and $v_{4}$ in the brown region of Figure 5). The
$v_{4}$ vacuum was described above (99), and $v_{3}^{-}$ does not differ much
from $v_{3}^{+}$: in fact, only the counter-terms for background fields
associated to the global symmetry (e.g. the metric) are going to be different.

The special case of $k_{1}>1$ and $k_{2}=1$, is pretty much similar, and the
resulting phase diagram is depicted in Figure 6. We note though that in this
case the large positive mass phase (pink region) does not have any
supersymmetric vacua, consequently there are just two vacua in the grey phase
and three vacua in the brown phase. It also implies that at the transition
line between the yellow region and the grey region two supersymmetric vacua
collide and, instead of producing a new supersymmetric vacuum, get lifted. In
the other special case, when $k_{2}=k_{1}+1$, in the $v_{2}$ vacuum
supersymmetry is broken and so the phase transition between the purple phase
and the light blue phase is absent.

Figure 6: Phase diagram of the $SU(2)_{k_{1}}\times SU(2)_{-1}$ quivers with
$k_{1}>1$. Dashed lines correspond to the second order phase transitions,
while the solid line is the wall. The supersymmetric vacua in each phase are
indicated.

We conclude this section with the following observation. While sitting exactly
at the wall and moving along it, we notice that at the flipping point for
$v_{3}$ (the second special point) the Witten index jumps (in fact, for
$k_{2}=1$ it vanishes below the flipping point, while being non-zero above).
This is consistent with the fact that an asymptotically flat direction opens
up at this point. Indeed, the transition can be arranged in the following way:
at the special point one vacuum goes away to infinity, while another vacuum
comes in from there. We thus conclude that exactly at the second flipping
point there are no supersymmetric vacua, and the model exhibits the runaway
behaviour (however, there might be meta-stable supersymmetry breaking vacua).
The first flipping point does not show any special behaviour for the
$SU(2)\times SU(2)$ quivers, but, as we will see below, there is an analogous
phase transition at the first flipping point for the $SU(2)\times U(2)$
quivers. We thus expect the following picture: when we move down along the
wall, at the first flipping point the supersymmetric vacuum goes away to
infinity, and after this point the identical vacuum comes in from infinity.
Again, at the special point the runaway behaviour takes place.

##  5 Phase diagrams of $SU(2)\times U(2)$ models

In this section we discuss the phases of $SU(2)\times U(2)$ quiver theories,
again with one bi-fundamental multiplet. Even though in principle one should
recompute the effective superpotential for this case, we will appeal to a
shortcut, and just assume that the vacuum structure (and in particular the
symmetry breaking patterns) are the same as we have seen before. The main
motivation for this assumption is that the Abelian factor inside U(2), for
large CS level, does not modify the behaviour of the vacua.

###  5.1 Chern-Simons levels of the same sign

We start by considering the models of the form

$\mathcal{N}=1\quad SU(2)_{k_{1}}\times U(2)_{k_{2},k_{3}}$ (100)

with the coupling to bi-fundamental matter, and we will restrict ourselves
with the case of $k_{1}\geq 0$, $k_{2}>1$.191919Consistency requires that
$k_{2}=k_{3}+1\,\text{mod}\,2$, such that the resulting IR TQFTs satisfy the
condition stated after (6).

####  5.1.1 Large mass asymptotic phases

As before, we can readily understand the large mass phases. When the mass is
large and positive, we get the IR theory

$SU(2)_{k_{1}}\times U(2)_{k_{2},k_{3}+1}\quad\text{TQFT}$ (101)

with the index202020In the case of $U(N)_{k_{2},k_{3}}$ the Witten index reads
$\text{WI}=\frac{(k_{2}+N-1)!\ k_{3}}{N!\ k_{2}!}$

$\text{WI}_{+}=-\frac{(k_{1}+1)(k_{2}+1)|k_{3}+1|}{2}.$ (102)

When the mass is large and negative, few different cases can be discussed:

  * •

When $k_{1}>1$ and $k_{2}>1$, there is one supersymmetric vacuum whose IR
theory is given by

$SU(2)_{k_{1}-2}\times U(2)_{k_{2}-2,k_{3}-1}\quad\text{TQFT}.$ (103)

The Witten index is

$\text{WI}_{-}=-\frac{(k_{1}-1)(k_{2}-1)|k_{3}-1|}{2}.$ (104)

  * •

When $k_{1}=1$, supersymmetry is spontaneously broken, and the IR theory is
given by

$G_{\alpha}+U(1)_{2}\times U(2)_{k_{2}-2,k_{3}-1}\quad\text{TQFT}.$ (105)

  * •

When $k_{1}=0$, we again see a supersymmetric vacuum hosting a CS theory,

$U(2)_{k_{2}-2,k_{3}-1}\quad\text{TQFT},$ (106)

and the index is

$\text{WI}_{-}=\frac{(k_{2}-1)|k_{3}-1|}{2}.$ (107)

Following the familiar strategy, it is then useful to understand the dynamics
near the wall, $m=0$, which we do now.

####  5.1.2 Non-Abelian vacuum

By assumption, there again exists a vacuum of the form

$\Phi=\left(\begin{matrix}v_{2}&&0\\\ 0&&v_{2}&\end{matrix}\right).$

The gauge group is still broken to $SU(2)$ with the induced CS level
$k_{1}+k_{2}$, but since the baryonic symmetry is now gauged, there are no
Goldstone modes in the IR. In fact, the would-be Goldstone boson superpartner
$\psi_{G}$ gets mixed with the $U(1)$ gaugino via the mass matrix

$\left(\begin{matrix}0&&-g_{2}v_{2}\\\
-g_{2}v_{2}&&-\kappa_{3}\end{matrix}\right).$ (108)

We assume that the rest of the fermionic spectrum is qualitatively the same,
and thus the IR theory is given by

$SU(2)_{k_{1}+k_{2}-2}\quad\text{TQFT}.$ (109)

There are ten negative-mass Majorana modes, so the Witten index is

$\text{WI}_{2}=k_{1}+k_{2}-1.$ (110)

####  5.1.3 Abelian vacuum

In the same way we expect to find a vacuum of the form

$\Phi=\left(\begin{matrix}v_{3}&&0\\\ 0&&0&\end{matrix}\right).$

It breaks the gauge group to $U(1)\times U(1)$, and the induced CS levels are
given by the matrix

$K=\left(\begin{matrix}2(k_{1}+k_{2})&&-k_{2}\\\
-k_{2}&&\frac{1}{2}(k_{2}+k_{3})\end{matrix}\right).$ (111)

We can now use the fermionic charges and masses computed in Section 4.1.2 to
obtain the quantum corrections to the level matrix induced upon the
integration out of the fermions:

$K_{IR}=\left(\begin{matrix}2(k_{1}+k_{2})&&-k_{2}\\\
-k_{2}&&\frac{1}{2}(k_{2}+k_{3})+\frac{1}{2}\end{matrix}\right).$ (112)

The Witten index of this vacuum is given (up to a sign) by the number of lines
in the corresponding Abelian CS theory,

$\text{WI}_{3}=\text{det}K_{IR}=-|(k_{1}+k_{2})(k_{2}+k_{3})+(k_{1}+k_{2})-k_{2}^{2}|.$
(113)

${\ast}\,{\ast}\,{\ast}$

The overall structure of the phase diagram is identical to the one depicted on
Figures (2),(3). We conjecture (following the pattern discussed in Section
4.1) that for $k_{3}\neq 0$ at the intermediate (yellow) phase there is still
the Abelian vacuum, as well as some other vacuum, resulting from the merging
of the Non-Abelian vacuum and the vacuum at the origin. This quantum vacuum is
expected to support a TQFT or/and a non-linear sigma model with the Witten
index fixed by the matching condition. When $k_{3}=1$, the vacuum structure
happens to be quite different, and will be discussed in Section 6.

###  5.2 Chern-Simons levels of the opposite signs

Next we discuss the models of the form

$\mathcal{N}=1\quad SU(2)_{k_{1}}\times U(2)_{-k_{2},-k_{3}}$ (114)

with, $k_{1}>k_{2}>1$.

####  5.2.1 Large mass asymptotic phases

When the matter mass is large and positive, we obtain

$SU(2)_{k_{1}}\times U(2)_{-k_{2}+2,-k_{3}+1}$ (115)

in the IR. There are three or four negative mass Majorana modes, depending on
whether $k_{3}$ is positive or negative, so the index is

$\text{WI}_{1}=-\frac{(k_{1}+1)(k_{2}-1)(k_{3}-1)}{2}.$ (116)

When instead the mass is large and negative, we find a supersymmetric vacuum
with

$SU(2)_{k_{1}-2}\times U(2)_{-k_{2},-k_{3}-1}\quad\text{TQFT},$ (117)
$\text{WI}_{1}=\text{sgn}(k_{3})\frac{(k_{1}-1)(k_{2}+1)|k_{3}+1|}{2}.$ (118)

####  5.2.2 Non-Abelian vacuum

Similarly to the $SU(2)\times SU(2)$ case, the non-Abelian vacuum is expected
to exist on the right from the wall for $g_{1}\gg g_{2}$, and on the left from
the wall for $g_{1}\ll g_{2}$, with a flipping point for some value of
$\frac{g_{1}}{g_{2}}$. The gauge group is broken to $SU(2)_{k_{1}-k_{2}}$. The
masses of fermions transforming in the adjoint representation don’t change
upon the crossing of the flipping point, there are always one of them with a
positive mass and three with negative masses. On the contrary, one of the
neutral Majorana fermions change the sign of its mass, such that there are
seven negative-mass Majorana modes when $m>0$ and eight negative-mass Majorana
modes when $m<0$. We therefore get in the IR

$SU(2)_{k_{1}-k_{2}-2}\quad\text{TQFT},\quad\text{WI}=-\text{sgn}(m)(k_{1}-k_{2}+1).$
(119)

####  5.2.3 Abelian vacuum

Finally, we suppose that there is an Abelian vacuum supporting the $U(1)\times
U(1)$ CS theory. This vacuum is also expected to flip from one side of the
wall to another at some value of $\frac{g_{1}}{g_{2}}$ (the second special
point), and fermions charged under the unbroken gauge group do not flip the
signs of their masses, and so the level matrix is given by

$K^{+}_{IR}=\left(\begin{matrix}2(k_{1}-k_{2})&&k_{2}\\\
k_{2}&&-\frac{1}{2}(k_{2}+k_{3})+\frac{1}{2}\end{matrix}\right).$ (120)

on both sides of the wall. Some neutral modes though flip their masses, so
that the Witten index is negative for $m>0$ and positive for $m<0$.

Decreasing the ratio $\frac{g_{1}}{g_{2}}$ even further, we expect to face the
third special point where a new Abelian vacuum with $B\neq 0$ appears ($v_{4}$
in the brown region of Figure 5). This vacuum preserves just one Abelian
factor, and supports

$U(1)_{2(k_{1}-k_{2})}\quad\text{TQFT}$ (121)

in the IR. The Abelian vacuum discussed above also undergoes some changes when
the third special point is passed. Namely, one of the fermions charged under
the second $U(1)$ gets a negative mass, which leads to the corrected level
matrix:

$K^{-}_{IR}=\left(\begin{matrix}2(k_{1}-k_{2})&&k_{2}\\\
k_{2}&&-\frac{1}{2}(k_{2}+k_{3})-\frac{1}{2}\end{matrix}\right).$ (122)

The Abelian vacuum with the $U(1)\times U(1)$ gauge group and the level matrix
given above is denoted by $v_{3}^{-}$ in Figure 5.

${\ast}\,{\ast}\,{\ast}$

The overall phase diagram looks quite similar to what is seen in Figures 5 and
6, despite it is now harder to guess the vacuum at the origin in yellow and
purple phases. We conjecture that for $k_{2}=2,k_{3}=1$ it is given by the
Abelian CS theory

$U(1)_{2k_{1}}.$ (123)

This conjecture is motivated by a duality discussed in the next section.
Slightly more generally, for $k_{3}=1$ it is natural to expect that this
vacuum supports the same TQFT in the IR, as does the Abelian vacuum also
existing in this phase, namely

$\left[U(1)\times U(1)\right]_{K}\quad\text{TQFT},$ (124)
$K=\left(\begin{matrix}2(k_{1}-k_{2})&&k_{2}\\\
k_{2}&&-\frac{1}{2}k_{2}\end{matrix}\right).$ (125)

The two proposal are consistent, since, as can be easily verified, for
$k_{2}=2$ $\left[U(1)\times U(1)\right]_{K}$ is dual to $U(1)_{2k_{1}}$

The last comment concerns the dynamics at the special points. Similarly to the
discussion at the end of Section 4.2, we observe two phase transitions, at the
first and at the second special points. As before, they are organized by first
sending a supersymmetric vacuum to infinity, and then receiving a new
supersymmetric vacuum, generically with a different TQFT and Witten index,
from infinity, with the runaway behaviour at the transition point.

##  6 Dualities

The discussion of the previous two sections demonstrated that a generic 3d
$\mathcal{N}=1$ quiver gauge theory has multiple second-order phase
transitions with associated IR fixed points. In this section we will provide
few conjectures stating that certain CFTs that appear as IR limits of
different quiver theories may in fact be the same: this is the statement of
the IR duality. Some of such dualities were already used above to guess
certain aspects of the phase diagrams (namely, the vacuum structures in the
intermediate phases).

###  6.1 Dualities between $SU(2)\times SU(2)$ and $SU(2)\times U(2)$

The first pair of theories we consider is $SU(2)_{k}\times U(2)_{2,1}$ and
$SU(2)_{k+2}\times SU(2)_{-2}$ quivers with $k>0$; the corresponding phase
diagrams are shown in Figure 7, and the yellow phase of 7 as well as the
yellow and light blue phases of 7 are conjectures. We observe, using the
level-rank duality

$SU(2)_{-2}\longleftrightarrow U(2)_{2,2},$ (126)

that the transition in Figure 7 between the yellow and the green phases is
identical to the transition in Figure 7 between the light blue phase and the
purple phase, with the phases on both sides of the transition given by

$\mathbb{S}^{1}\times SU(2)_{k-2}+SU(2)_{k}\times SU(2)_{-2}\longrightarrow
SU(2)_{k}\times SU(2)_{-2}.$ (127)

While making the conjecture about the vacua in the yellow phase of Figure 7,
we assumed that at the transition point $m=m_{*}$ the non-Abelian vacuum
merges with the Abelian one, while the vacuum at the origin stays apart. This
is in contrast with what was assumed in Sections 4.1 and 5.1. The difference
comes from the fact that here a new ”branch” of vacua, parametrized by the
dual photon, emerges.

The space of possible vacua can be then visualized as the two-dimensional
space parametrized by the scalar vevs, together with a cone attached at the
origin (see the Figure on the right). The angular direction of the cone is the
dual photon, and the radial direction gives the radius of the circle (which is
not a dynamical field, but rather a function of the parameters). It is then
possible that first the non-Abelian and the Abelian vacua meet at the origin,
and then the dual photon radius (as a function of $m$) shrinks to zero, and
the second phase transition happens.

Figure 7: Phase diagram for the $SU(2)_{k}\times U(2)_{2,1}$ quiver (a) and
for the $SU(2)_{k+2}\times SU(2)_{-2}$ quiver (b).

The second pair is $SU(2)_{k}\times SU(2)_{2}$ and $SU(2)_{k+2}\times
U(2)_{-2,-1}$ quivers with $k>0$; the corresponding phase diagrams are
presented in Figure 8, and the yellow phase of Figure 8 is a conjecture. We
propose that the ”quantum” vacuum $v_{q}$ in Figure 8 is given by

$U(1)_{2(k+2)}\quad\text{TQFT}.$ (128)

Figure 8: Phase diagram for the $SU(2)_{k}\times SU(2)_{0}$ quiver (a) and for
the $SU(2)_{k}$ adjoint QCD (b).

Using again the level-rank duality (126), we again observe that the transition
in Figure 8 between the yellow and the green phases is identical to the
transition in Figure 8 between the light blue phase and the purple phase,

$SU(2)_{k-2}+U(1)_{2(k+2)}\longrightarrow SU(2)_{k}\times SU(2)_{2}.$ (129)

The two dualities we have just described can be obtained from the duality (13)
by gauging the flavour $SU(2)$ (sub)groups on both sides.

###  6.2 Duality between $SU(2)\times SU(2)$ quiver and adjoint QCD

The first model considered here is the $SU(2)_{k}\times SU(2)_{0}$ quiver,
discussed in section 4.1.2. The phase diagram can be found in Figure 9, where
the form of the yellow phase is a conjecture. The phases of the $SU(2)_{k}$
adjoint QCD were reviewed in Section 2, and are depicted on Figure 9.

Figure 9: Phase diagram for the $SU(2)_{k}\times SU(2)_{0}$ quiver (a) and for
the $SU(2)_{k}$ adjoint QCD (b).

Evidently, the phase transitions between the yellow phases and the green
phases are identical, and this hints towards the possibility of the duality:

$SU(2)_{k}\times SU(2)_{0}\quad\text{with a bi-
fundamental}\quad\longleftrightarrow\quad SU(2)_{k}\quad\text{with an
adjoint}.$ (130)

This duality if correct has a quite clear meaning. Assuming that the
$SU(2)_{0}$ node of the quiver confines, we can describe the low-energy
physics in terms of the bilinears

$X=\Phi\Phi^{\dagger},$ (131)

which indeed transforms in the adjoint representation of $SU(2)_{k}$. There is
one point in this picture that may seem disturbing. The quiver theory
possesses the baryonic symmetry $U(1)_{B}$, and there are charged operators
$B=\det\Phi$. Neither the symmetry, nor would-be dual operators appear on the
QCD side. This issue can be resolved in two ways: either the quiver theory
baryons happen to be massive, and do not appear in the IR fixed point, or they
are actually massless at the CFT point, but decouple from the rest. In the
latter case the QCD side should be supplemented by a decoupled free complex
multiplet.

##  7 Time reversal invariant models

We have already mentioned that $3d$ $\mathcal{N}=1$ theories with time
reversal invariance have a beautiful property: their superpotentials admit
only corrections odd under the action of $T$-transformation Gaiotto:2018yjh .
It significantly restricts the possible form of the effective superpotential,
and sometimes superpotential even turns out to be fully protected.

Examples of $T$-invariant theories can be found also among the quiver
theories. For example, a two-node quiver with opposite CS levels,

$\mathcal{N}=1\quad SU(2)_{k}\times SU(2)_{-k}\,+\,\text{a bi-fundamental},$
(132)

enjoys this property at the point $g_{1}=g_{2}$, $m=0$212121To be more
precise, $T$-transformation must be augmented by the exchange of two gauge
group factors.. It is easy to see that there are no parity odd terms that
could be written in the effective superpotential, implying that we have an
example with full protection at hands. In fact, one can check that the 1-loop
superpotential computed in Section 3 vanishes at this point. It follows that
the theory has a moduli space of vacua, which coincides with the classical
one,

$\mathcal{M}=\mathbb{S}^{1}\,\times\,\mathbb{R}^{2}\,/\,S_{2}.$ (133)

At the origin of the moduli space we expect to find a SCFT. At a point away
from the origin the IR physics is described by three real massless moduli
without any topological sector.

We can then deform the theory from the $T$-invariant point by turning on the
mass term or changing the ratio $\frac{g_{1}}{g_{2}}$ and study the resulting
IR phases. The large mass phases for $k>1$ are supersymmetric and are given by

$SU(2)_{k}\times SU(2)_{-k+2}\quad\text{TQFT}$ (134)

for large positive masses and

$SU(2)_{k-2}\times SU(2)_{-k}\quad\text{TQFT}$ (135)

for large negative masses. When $k=1$, supersymmetry is spontaneously broken,
and the IR description is

$G_{\alpha}+U(1)_{2}\times U(1)_{2}\quad\text{TQFT}$ (136)

for large positive masses and the same for large positive masses, where the
level-rank duality (83) has been used.

Figure 10: Structure of the phase diagram for the $SU(2)_{k}\times SU(2)_{-k}$
quiver. Dashed lines correspond to the second order phase transitions, while
the solid line is the wall. The supersymmetric vacua in each phase are
indicated.

We can also study behaviour near the wall and find that for
$\frac{g_{1}}{g_{2}}>1$ to new vacua appear when $m>0$, while for
$\frac{g_{1}}{g_{2}}<1$ we find them when $m<0$. These are would-br Abelian
and non-Abelian vacua familiar from above, dipite the fact that they do not
support any topological degrees of freedom. Still, they are not trivial and
host $S^{1}$ Goldstone bosons: Abelian vacuum breaks spontaneously magnetic
symmetry of the preserved $U(1)$ gauge group, while non-Abelian vacuum breaks
the baryonic symmetry. The phase diagram is depicted on Figure 10.

##  8 Outlook

In this paper the study of $3d$ $\mathcal{N}=1$ quiver theories was initiated.
Already the simplest possible setup of a two-node quiver with $SU(2)$ gauge
groups and one bi-fundamental multiplet reveals quite rich and diverse
physical pictures. We find the characteristic features of theories with two
supercharges observed previously in the literature: walls in the parameter
space at which the Witten index jumps, multiple phases with second order phase
transitions between them, vacua with spontaneously broken supersymmetry, which
can be either stable or meta-stable. Especially interesting phase diagrams are
found in theories with CS levels of different signs (Sections 4.2, 5.2). The
study of phase diagrams was facilitated by some duality considerations:
conjecturing certain dualities, one sometimes could gain more understanding of
the intermediate ”quantum” phases, which are not continuously connected to any
weakly coupled limit.

Two of the dualities considered in the paper can be related to previously
conjectured dualities for SQCD-like theories, and obtained from them by
gauging flavour symmetries on both sides. This tool for generating new
dualities is well-known for theories with greater amount of supersymmetry, but
our results suggest that it is also applicable in the realm of minimally
supersymmetric three-dimensional theories. The third duality we discuss
involves the confinement of a node, and corresponds to the situation for which
at low energies the physics can be described in terms of bilinears (or more
generally multi-linears) with respect to the original matter field, and gauge-
neutral with respect to a given node.

Many results of this paper offer directions for generalizations. The most
obvious one is to consider quivers with higher-rank gauge groups, as $SU(N)$
or $U(N)$. The overall structure of the phase diagrams is expected to be more
complicated in these cases but nevertheless one may hope to understand the
dynamics near the wall (even though the 1-loop superpotential is going to have
a more complicated structure). More general dualities of Choi:2018ohn  can
then be used, together with the node-dualization technique, to conjecture new
dualities between quivers. In this case, a more detailed analysis is required
in order to establish to which of the multiple SCFTs the duality applies.

Another interesting generalization is to take several bi-fundamental
multiplets. The resulting vacua will generically break flavour symmetries
therefore the IR description will be given in terms of non-linear sigma
models, in addition to the topological sectors already seen above.

We have already mentioned in the introduction that simple quiver theories,
like the ones that we have been studying in this work, are prototypical
examples of more complicated models arising as world-volume theories on M2
branes put on certain backgrounds Forcella:2009jj . Since classically these
theories have moduli spaces, it is natural to think that they appear exactly
on the wall inside their phase diagrams. Borrowing some lessons from the
results we found above, we known that at quantum level different possibilities
can arise: the vacuum may be supersymmetric and host some topological sector,
or it may break supersymmetry, or show a runaway behaviour with asymptotically
flat direction. It is extremely interesting to investigate which of these
possibilities are realized in various string/M-theory configurations.

To conclude, there remain many exciting questions regarding the dynamics of
three-dimensional quivers with minimal sypersymmetry, and we hope to report on
the progress in some of them in the future.

## Acknowledgements

We thank Joseph Minahan, Adar Sharon, and Zohar Komargodski for valuable
discussions. The work of VB is supported by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No. 851931). The work of NG is supported by the
“Gauge Theories, Strings, Supergravity” (GSS) research project.

##  Appendix A Group Theory Conventions

The gauge group we consider in this paper is of the form $G=G_{1}\times
G_{2}$. In the main text $G_{1,2}$ are chosen to be $SU(2)$ so we restrict to
this one.
The $\mathfrak{su}(2)$ Lie algebra is defined through the usual relation

$[\tau^{A},\tau^{B}]=i\epsilon^{ABC}\tau^{C}$ (137)

where $A,B,C$ are adjoint indices. We choose
$\tau^{A}\equiv\frac{1}{2}\sigma^{A}$, where $\sigma^{A}$ are the Pauli
matrices, so that the canonical normalization

$\text{Tr}\left(\tau^{A}\tau^{B}\right)=\frac{1}{2}\delta^{AB}$ (138)

holds. We make also use of the following notation

$\tau^{(A}\tau^{B)}\equiv\frac{1}{2}\\{\tau^{A},\tau^{B}\\},\qquad\tau^{[A}\tau^{B]}\equiv\frac{1}{2}[\tau^{A},\tau^{B}]$
(139)

Since we are considering a quiver gauge theory with the matter sitting in the
(anti-)bifundamental representation of $G_{1}\times G_{2}$, namely
$(\bar{R},R)$ and $(R,\bar{R})$ respectively, we recall the action of $G$ on
them. The gauge group transformations $U\in G_{1}$ and $V\in G_{2}$ will then
act in the following way

$(\Phi^{\prime})_{i}^{\ \hat{j}}=(U\Phi V^{\dagger})_{i}^{\
\hat{j}},\qquad(\bar{\Phi}^{\prime})_{\hat{i}}^{\
j}=(V\bar{\Phi}U^{\dagger})_{\hat{i}}^{\ j}$ (140)

and at the level of algebra we have

$\delta^{(A)}\Phi_{i}^{\ \hat{j}}=i\left[g_{1}(T^{(A)})_{i}^{\ k}\Phi_{k}^{\
\hat{j}}-g_{2}\Phi_{i}^{\ \hat{k}}(K^{(A)})_{\hat{k}}^{\
\hat{j}}\right],\qquad\delta^{(A)}\bar{\Phi}_{\hat{i}}^{\
j}=i\left[g_{2}(K^{(A)})_{\hat{i}}^{\ \hat{k}}\bar{\Phi}_{\hat{k}}^{\
j}-g_{1}\Phi_{\hat{i}}^{\ k}(T^{(A)})_{k}^{\ j}\right]$ (141)

where $g_{1,2}$ are the two gauge couplings for $G_{1,2}$ and $T^{(A)}$,
$K^{(A)}$ are generators of the $\mathfrak{g}_{1},\mathfrak{g}_{2}$ Lie
algebras respectively.

##  Appendix B Superspace Conventions

The conventions we used in the main text refer to Gates:1983nr ; Choi:2018ohn
.
We raise and lower spinor indices through the use of the antisymmetric matrix
$C_{\alpha\beta}$ as follows

$\psi^{\alpha}=C^{\alpha\beta}\psi_{\beta},\quad\psi_{\alpha}=\psi^{\beta}C_{\beta\alpha}=-C_{\alpha\beta}\psi^{\beta}$
(142)

with $\psi^{2}\equiv\frac{1}{2}\psi^{\alpha}\psi_{\alpha}$. The graded
commutation relation for derivatives is

$\\{D_{\alpha},D_{\beta}\\}=2i\partial_{\alpha\beta}$ (143)

where $\partial_{\alpha\beta}$ is the ordinary spacetime derivative.
Derivatives also satisfy the following useful identities

$\displaystyle\qquad\qquad\partial^{\alpha\gamma}\partial_{\beta\gamma}=\delta^{\alpha}_{\beta}\Box,\quad
D_{\alpha}D_{\beta}=i\partial_{\alpha\beta}+C_{\alpha\beta}D^{2}$ (144)
$\displaystyle D_{\alpha}D_{\beta}D^{\alpha}=0,\quad
D^{2}D_{\alpha}=-D_{\alpha}D^{2}=i\partial_{\alpha\beta}D^{\beta},\quad(D^{2})^{2}=\Box$

###  B.1 $\mathcal{N}=1$ Gauge Theories

The most generic kinetic terms for three-dimensional supersymmetric gauge
theories are Chern-Simons and Yang-Mills ones. In the $\mathcal{N}=1$
notation, the Lagrangian for such terms reads

$\mathcal{L}_{\text{CS-
YM}}=-\frac{k}{4\pi}\Tr(2i\Gamma^{\alpha}\partial_{\alpha\beta}\Gamma^{\beta}+\Gamma^{\alpha}D_{\alpha}D^{\beta}\Gamma_{\beta})+\frac{1}{2g^{2}}\Tr(\Gamma^{\alpha}\Box\Gamma_{\alpha}-i\Gamma^{\alpha}\partial_{\alpha\beta}D^{2}\Gamma^{\beta})$
(145)

which, with some effort, can be recasted in the following gauge-fixed form

$\displaystyle\mathcal{L}_{\text{CS-YM}}^{\text{gf}}=$
$\displaystyle-\frac{k}{4\pi}\Tr(2i\Gamma^{\alpha}\partial_{\alpha\beta}\Gamma^{\beta}+\left(1-\frac{1}{\beta}\right)\Gamma^{\alpha}D_{\alpha}D^{\beta}\Gamma_{\beta})$
(146)
$\displaystyle+\frac{1}{2g^{2}}\Tr(\left(1+\frac{1}{\alpha}\right)\Gamma^{\alpha}\Box\Gamma_{\alpha}-i\left(1-\frac{1}{\alpha}\right)\Gamma^{\alpha}\partial_{\alpha\beta}D^{2}\Gamma^{\beta})$

By taking the Landau gauge-fixing limit ($\alpha,\beta\rightarrow 0$) one can
obtain the final form for the CS-YM gauge propagator, which reads

$\Delta_{\alpha}^{\ \beta}=g^{2}\frac{\delta_{\alpha}^{\beta}(\kappa
D^{2}+p^{2})+(\kappa-D^{2})p_{\alpha}^{\
\beta}}{p^{2}(\kappa^{2}+p^{2})},\qquad\kappa=\frac{kg^{2}}{2\pi}$ (147)

Matter can be also coupled through the following action

$S_{\text{matter}}=-\frac{1}{2}\int d^{3}xd^{2}\theta\
(\nabla^{\alpha}\bar{\Phi})(\nabla_{\alpha}\Phi)$ (148)

where in our setup, the covariant derivatives take the following explicit form

$\displaystyle\nabla_{\alpha}{\Phi_{i}}^{\hat{j}}$
$\displaystyle=D_{\alpha}{\Phi_{i}}^{\hat{j}}-ig_{1}\
\Gamma^{A}_{\alpha}{(T^{A})_{i}}^{k}{\Phi_{k}}^{\hat{j}}+ig_{2}\
\hat{\Gamma}^{M}_{\alpha}\ {\Phi_{i}}^{\hat{k}}{(K^{M})_{\hat{k}}}^{\hat{j}}$
(149) $\displaystyle\nabla_{\alpha}{{\bar{\Phi}}_{\hat{j}}}^{\ i}$
$\displaystyle=D_{\alpha}{\bar{\Phi}_{\hat{j}}}^{\ i}-ig_{2}\
\hat{\Gamma}^{M}_{\alpha}{(K^{M})_{\hat{j}}}^{\hat{k}}{\bar{\Phi}_{\hat{k}}}^{\
i}+ig_{1}\ \Gamma^{A}_{\alpha}\ {\bar{\Phi}_{\hat{j}}}^{\ k}{(T^{A})_{k}}^{i}$
(150)

##  Appendix C Effective superpotential at $k_{1}=-k_{2}$

When the CS levels are equal by the absolute value but opposite, one can
compute derivatives of the superpotential explicitely. Introducing
$k_{1}=-k_{2}=k$, $\kappa_{1}=\frac{kg1^{2}}{2\pi}$,
$\kappa_{2}=\frac{kg1^{2}}{2\pi}$, we get for $B\rightarrow\rho$:

$\displaystyle\partial_{\rho}\mathcal{W}_{\text{1-loop}}=-\frac{(g_{1}^{2}+g_{2}^{2})(\kappa_{1}-\kappa_{2})\left[(9g_{1}^{4}+22g_{1}^{2}g_{2}^{2}+9g_{2}^{4})\rho+6(g_{1}^{2}+g_{2}^{2})(\kappa_{1}^{2}+\kappa_{1}\kappa_{2}+\kappa_{2}^{2})\right]}{16\pi(2\kappa_{1}\kappa_{2}+\rho(g_{1}^{2}+g_{2}^{2}))\sqrt{(k_{1}+k_{2})^{2}+2\rho(g_{1}^{2}+g_{2}^{2})}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\overset{\rho\rightarrow\infty}{\longrightarrow}-\frac{(\kappa_{1}-\kappa_{2})(9g_{1}^{4}+22g_{1}^{2}g_{2}^{2}+9g_{2}^{4})}{16\pi\sqrt{2}\sqrt{(g_{1}^{2}+g_{2}^{2})\rho^{1/2}}},$
(151a)
$\displaystyle\partial_{|B|}\mathcal{W}_{\text{1-loop}}=\frac{g_{1}^{2}g_{2}^{2}(\kappa_{1}-\kappa_{2})\rho}{4\pi(2\kappa_{1}\kappa_{2}+\rho(g_{1}^{2}+g_{2}^{2}))\sqrt{(k_{1}+k_{2})^{2}+2\rho(g_{1}^{2}+g_{2}^{2})}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\overset{\rho\rightarrow\infty}{\longrightarrow}\frac{g_{1}^{2}g_{2}^{2}(\kappa_{1}-\kappa_{2})}{4\sqrt{2}\pi(g_{1}^{2}+g_{2}^{2})^{3/2}\rho^{1/2}},$
(151b)

and in the limit $B\rightarrow 0$ the result is

$\displaystyle\partial_{\rho}\mathcal{W}_{\text{1-loop}}=-\frac{\kappa_{1}-\kappa_{2}}{8\pi(g_{1}^{2}-g_{2}^{2})^{2}D}\left[\frac{N_{1}}{\sqrt{\kappa_{1}^{2}+2g_{1}^{2}\rho}}+\frac{N_{2}}{\sqrt{\kappa_{2}^{2}+2g_{2}^{2}\rho}}\right]-$
$\displaystyle\quad-\frac{(g_{1}^{2}+g_{2}^{2})(\kappa_{1}-\kappa_{2})(2\kappa_{1}^{2}+2\kappa_{1}\kappa_{2}+2\kappa_{2}^{2}+3\rho(g_{1}^{2}+g_{2}^{2}))}{16\pi(2\kappa_{1}\kappa_{2}+\rho(g_{1}^{2}+g_{2}^{2}))\sqrt{(\kappa_{1}+\kappa_{2})^{2}+2\rho(g_{1}^{2}+g_{2}^{2})}}$
$\displaystyle\qquad\qquad\qquad\overset{\rho\rightarrow\infty}{\longrightarrow}-\frac{g_{1}\kappa_{1}^{2}-g_{2}\kappa_{2}^{2}}{8\sqrt{2}\pi}-\frac{\kappa_{1}^{2}g_{1}^{2}-\kappa_{2}^{2}g_{2}^{2}}{16\sqrt{2}\pi\sqrt{g_{1}^{2}+g_{2}^{2}}\rho^{1/2}},$
(152a)
$\displaystyle\partial_{|B|}\mathcal{W}_{\text{1-loop}}=\frac{g_{1}^{2}g_{2}^{2}(\kappa_{1}-\kappa_{2})B}{4\pi(g_{1}^{2}-g_{2}^{2})D}\left[\frac{g_{1}^{4}+3g_{1}^{2}g_{2}^{2}}{\sqrt{\kappa_{1}^{2}+2g_{1}^{2}\rho}}-\frac{g_{2}^{4}+3g_{1}^{2}g_{2}^{2}}{\sqrt{\kappa_{2}^{2}+2g_{2}^{2}\rho}}\right]$
$\displaystyle\qquad\qquad\qquad\overset{\rho\rightarrow\infty}{\longrightarrow}\frac{g_{1}g_{2}(\kappa_{1}g_{2}-\kappa_{2}g_{2})}{4\sqrt{2}\pi(g_{1}+g_{2})^{2}}.$
(152b)

where

$\displaystyle
D=(g_{1}^{2}-g_{2}^{2})^{2}\rho-4(\kappa_{1}^{2}g_{2}^{2}+\kappa_{2}^{2}g_{1}^{2}).$
(153a) $\displaystyle
N_{1}=g_{1}^{4}(g_{1}^{2}-g_{2}^{2})^{3}\rho-4g_{1}^{8}\kappa_{2}^{2}+4g_{1}^{2}g_{2}^{2}\kappa_{1}^{2}(-g_{1}^{4}+g_{1}^{2}g_{2}^{2}+g_{2}^{4}),$
(153b) $\displaystyle
N_{2}=g_{2}^{4}(g_{2}^{2}-g_{1}^{2})^{3}\rho-4g_{2}^{8}\kappa_{1}^{2}+4g_{2}^{2}g_{1}^{2}\kappa_{2}^{2}(-g_{2}^{4}+g_{1}^{2}g_{2}^{2}+g_{1}^{4}).$
(153c)

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