††institutetext: Centre for Particle Theory, Department of Mathematical
Sciences, Durham University, South Road, Durham DH1 3LE, UK

#  $2$-index chiral gauge theories

Mohamed M. Anber  Samson Y.L. Chan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

W

e undertake a systematic study of the $4$-dimensional $SU(N)$ $2$-index chiral
gauge theories and investigate their faithful global symmetries and dynamics.
These are a finite set of theories with fermions in the $2$-index symmetric
and anti-symmetric representations, with no fundamentals, and they do not
admit a large-$N$ limit. We employ a combination of perturbative and
nonperturbative methods, enabling us to constrain their infrared (IR) phases.
Specifically, we leverage the ’t Hooft anomalies associated with continuous
and discrete groups to eliminate a few scenarios. In some cases, the anomalies
rule out the possibility of fermion composites. In other cases, the interplay
between the continuous and discrete anomalies leads to multiple higher-order
condensates, which inevitably form to match the anomalies. Further, we
pinpoint the most probable symmetry-breaking patterns by searching for
condensates that match the full set of anomalies resulting in the smallest
number of IR degrees of freedom. Higher-loop $\beta$-function analysis
suggests that a few theories may flow to a conformal fixed point.

##  1 Introduction

Chiral gauge theories form the fundamental framework of the Standard Model
(SM) of particle physics. Within the SM, the electroweak sector undergoes
Higgsing at weak coupling, allowing us to apply perturbative techniques.
However, without a Higgs field, gauge theories generally flow towards a
strongly-coupled regime, rendering their study considerably more challenging.
A non-comprehensive list of some of the recent papers that studied chiral
gauge theories is Bolognesi:2022beq ; Bolognesi:2021yni ; Bolognesi:2020mpe ;
Bolognesi:2017pek ; Bai:2021tgl ; Csaki:2021xhi ; Csaki:2021aqv ;
Anber:2021iip ; Smith:2021vbf ; Sheu:2022odl .

This paper focuses on a class of $SU(N)$ chiral gauge theories that
accommodate fermions in the $2$-index symmetric and anti-symmetric
representations. These theories, referred to as $2$-index chiral gauge
theories, can be characterized by the pair $(N,k)$, where $N$ represents the
color and $k$ serves as a common divisor of $N+4$ and $N-4$. Moreover, $k$ is
directly associated with the number of flavors in the $2$-index symmetric and
anti-symmetric representations. What makes these theories particularly
intriguing is the absence of a requirement to introduce fundamental fermions
to cancel the gauge anomaly. Additionally, they exhibit non-asymptotic freedom
for $N>44$. This class encompasses a collection of $14$ distinct theories,
occupying a distinct region within asymptotically-free chiral gauge theories.
Consequently, a systematic approach to studying this class is justified. It is
divided into two subclasses: bosonic and fermion theories. The latter can
accommodate gauge-invariant massless fermions. In comparison, the gauge-
invariant operators in bosonic theories cannot have a spinor index, as the
fermion number is gauged.

We initiated this study in Anber:2021iip , utilizing ’t Hooft anomalies to
constrain the infrared dynamics of two theories. Namely, these are $(N=8,k=4)$
and $(N=8,k=2)$ theories. One important development was the identification of
the faithful global symmetry acting on fermions. This enabled us to turn on
the most general discrete fluxes, the color-flavor-$U(1)$ (CFU) fluxes
compatible with the theory and, thus, utilize the full power of ’t Hooft
anomaly matching conditions. These anomalies are dubbed CFU anomalies. A
theory with ’t Hooft anomalies cannot be trivially gapped; the infrared (IR)
spectrum must contain massless particles or multi-vacua. In the case
$(N=8,k=4)$, we found that the condensation of two operators can saturate the
anomalies.

Continuing our explorations within this comprehensive framework, our current
investigation exhausts all the $14$ theories and introduces a few novel
aspects.

  1. 1.

We incorporate anomalies stemming from discrete symmetries, thereby imposing
additional constraints on the infrared spectra. In the context of the
$2$-index chiral gauge theories we are examining, in addition to continuous
non-abelian flavor symmetries, an axial $U(1)_{A}$ symmetry comes into play.
When a bosonic operator condenses, it generally breaks the $U(1)_{A}$ symmetry
down to a discrete subgroup, which typically is anomalous. Consequently, we
face the challenge of identifying a set of condensates that not only matches
the anomalies associated with non-abelian symmetries but also avoids the
presence of any anomalous unbroken discrete subgroups. For example, the two
candidate condensates we previously considered in the case $(N=8,k=4)$,
Anber:2021iip , fail to match the anomaly of an unbroken discrete group. We
revise the situation in light of the new understanding and propose that the
set of anomalies can be matched by other condensates.

Interestingly, in a few cases, matching the full set of anomalies,
particularly anomalies of discrete symmetries, can be achieved only via the
condensation of multiple higher-order operators. Given the strong dynamics,
such formation is not a surprise. However, anomalies explain the kinematical
reasons why such condensates have to form.

  2. 2.

Another significant aspect of our work lies in our pursuit of the minimal
scenario that satisfies the entire set of anomalies and yields the smallest
number of massless particles in the infrared spectrum. Such a scenario holds
particular appeal as it minimizes the free energy associated with the theory.

  3. 3.

We adhered to a systematic algorithm during our quest to identify composite
massless fermions capable of satisfying the anomalies within the fermionic
class. Regrettably, we were unable to find such composites. Notably, in the
case of $(N=6,k=2)$, we demonstrate that these composites cannot solely match
the CFU anomaly. Consequently, we are left with two plausible explanations:
either these composites do not exist altogether, or the formation of
condensates alongside the composites is necessary to match the anomaly.
However, the latter scenario is rather contrived in that it requires the
formation of special condensates (that do not alter the anomalies matched by
the composites) in addition to the composite fermions, prompting us to lean
toward the likelihood that composites cannot form in this case.

  4. 4.

To complete our study, we also examined the possibility that a theory flows to
a conformal fixed point in the IR. Generally speaking, a theory can form a
strongly-coupled IR fixed point, which is beyond the scope of any perturbative
analysis. Such a fixed point automatically satisfies the full set of
anomalies, albeit it remains an open question how this can be seen.
Nevertheless, by scrutinizing the higher-order $\beta$-function, we
successfully identified several instances where the analysis of the
$\beta$-function offers indications suggestive of a perturbative nature
inherent to a fixed point.

This paper is organized as follows. In Section 2, we review the global
symmetries and anomalies of the $2$-index chiral theories. This includes the
anomalies of continuous symmetries, the CFU anomalies, as well as anomalies of
discrete symmetries. In Section 3, we revise the matching of the CFU anomalies
in the IR and introduce the novelty of matching the discrete subgroups of the
axial $U(1)_{A}$ that can be left unbroken by a condensate. Sections 4 and 5
are devoted to applying these ideas to both the fermionic and bosonic field
theories, respectively. We summarize our findings in Section 6, and in
particular, the reader is referred to Table 10, which, for all theories, it
gives the global symmetries, the proposed IR condensates that yield the
smallest number of Goldstones, and the fate of the symmetries in the IR.

##  2 Theory: symmetry structure and anomalies

We consider $SU(N)$ gauge theory with $n_{\psi}$ flavors of left-handed Weyl
fermions $\psi$ transforming in the $2$-index representation along with
$n_{\chi}$ flavors of left-handed Weyl fermions $\chi$ transforming in the
complex conjugate $2$-index anti-symmetric representation:

$\displaystyle{\cal L}=-\frac{1}{2g^{2}}\mbox{tr}[f^{c}\wedge\star
f^{c}]-i\bar{\psi}\bar{\sigma}^{\mu}D_{\mu}\psi-i\bar{\chi}\bar{\sigma}^{\mu}D_{\mu}\chi\,,$
(1)

where $D_{\mu}\equiv\partial_{\mu}-ia_{\mu}^{c}$ is the covariant derivative,
$a^{c}$ is the color gauge field, and $f^{c}$ is its field strength. In this
work, we use the lower-case letters, $a^{c}$ and $f^{c}=da^{c}$, to denote the
dynamical (color) $1$-form gauge field and its field strength, while we use
upper-case letters, $A$ and $F=dA$, for background fields. To keep track of
the color indices, we choose $\psi_{(a_{1}a_{2})}$ to carry two down indices,
while $\chi_{[a_{1}a_{2}...a_{N-2}]}$ carries $N-2$ down indices. A round
(square) bracket indicates symmetrizing (anti-symmetrizing) over the indices.
The cubic anomaly coefficients of the $2$-index symmetric and the conjugate of
the $2$-index anti-symmetric representations are ${\cal A}_{\psi}=N+4$, ${\cal
A}_{\chi}=-(N-4)$, respectively. Cancellation of the gauge anomaly demands
that $n_{\psi}$ and $n_{\chi}$ are fixed as

$\displaystyle n_{\psi}=\frac{N-4}{k}\,,\quad n_{\chi}=\frac{N+4}{k}\,,$ (2)

where $k$ is a common divisor of $N-4$ and $N+4$. The theory is asymptotically
free provided that $11N-\frac{2(N^{2}-8)}{k}>0$. This leaves us with the
finite set of theories in Table 1. These theories do not possess a large-$N$
limit, as they become infrared-free for $N>44$. Also, except for $N=5,6,10$,
the other allowed colors are multiples of $4$. These are bosonic theories
because all their gauge invariant operators cannot carry a spinor index. In
other words, the $(-1)^{F}$ fermion number in bosonic theories is gauged, and
thus, they cannot have gauge-singlet fermionic operators.

One important aspect of this work is to systematically analyze these theories,
paying particular attention to the faithful global symmetries, and exhausting
the class of generalized ’t Hooft anomalies that enable us to constrain the
infrared phases.

$N$ | $5$ | $6$ | $8$ | $10$ | $12$ | $16$ | $20$ | $28$ | $36$ | $44$
---|---|---|---|---|---|---|---|---|---|---
$k$ | $1$ | $1,2$ | $2,4$ | $2$ | $4,8$ | $4$ | $4,8$ | $8$ | $8$ | $8$

.

Table 1: A list of the $2$-index chiral gauge theories.

###  2.1 Symmetries

The theory enjoys two global flavor groups $SU(n_{\psi})=SU((N-4)/k)$ and
$SU(n_{\chi})=SU((N+4)/k)$ acting on $\psi$ and $\chi$, respectively. In
addition, the theory is endowed with two $U(1)$ global classical symmetries,
$U(1)_{1}\times U(1)_{2}$. Their action on $\psi$ and $\chi$ is chosen as

$\displaystyle U(1)_{1}:\quad\psi\longrightarrow
e^{i\alpha_{1}}\psi\,,\quad\chi\longrightarrow e^{i\beta_{1}}\chi\,,$
$\displaystyle U(1)_{2}:\quad\psi\longrightarrow
e^{i\alpha_{2}}\psi\,,\quad\chi\longrightarrow e^{i\beta_{2}}\chi\,.$ (3)

The two transformations $U(1)_{1}$ and $U(1)_{2}$ come naturally with two
parameters. Here, however, we introduce the $4$ parameters $\alpha_{1,2}$ and
$\beta_{1,2}$ to account for the fermions charges, in addition to the
transformation parameters. The gauge sector instantons break most of the
classical $U(1)$ symmetries. The effective action in the instanton background
acquires the terms

$\displaystyle\Delta S$ $\displaystyle=$ $\displaystyle
i\left(n_{\psi}\alpha_{1}T_{\psi}+n_{\chi}\beta_{1}T_{\chi}\right)\int_{\mathbb{M}^{4}}\lambda_{0}^{u_{1}}\frac{\mbox{tr}\left[f^{c}\wedge
f^{c}\right]}{8\pi^{2}}+i\left(n_{\psi}\alpha_{2}T_{\psi}+n_{\chi}\beta_{2}T_{\chi}\right)\int_{\mathbb{M}^{4}}\lambda_{0}^{u_{2}}\frac{\mbox{tr}\left[f^{c}\wedge
f^{c}\right]}{8\pi^{2}}$

upon performing simultaneous transformations of $U(1)_{1}\times U(1)_{2}$,
where

$\displaystyle T_{\psi}=N+2\,,\quad T_{\chi}=N-2$ (5)

are the Dynkin indices of the representations. Here, $f^{c}$ is the $2$-form
field strength of the color group, while $\lambda^{u_{1}}$ and
$\lambda^{u_{2}}$ are the gauge parameters of $U(1)_{1}$ and $U(2)_{2}$,
respectively, i.e., $A^{u_{1,2}}\longrightarrow
A^{u_{1,2}}+d\lambda^{u_{1,2}}$. We can find a combination of the parameters
$\alpha_{1}$ and $\beta_{1}$ that kills the first term in $\Delta S$, leaving
behind a genuine symmetry. We call this symmetry the axial $U(1)_{A}$. It acts
on $\psi$ and $\chi$ with transformation parameter $\alpha$:

$\displaystyle U(1)_{A}:\quad\psi\longrightarrow e^{i2\pi\alpha
q_{\psi}}\psi\,,\quad\chi\longrightarrow e^{i2\pi\alpha q_{\chi}}\chi\,,$ (6)

and we have defined the $U(1)_{A}$ charges of $\psi$ and $\chi$ as

$\displaystyle q_{\psi}\equiv-\frac{N_{\chi}}{r}\,,\quad
q_{\chi}\equiv\frac{N_{\psi}}{r}\,,$ (7)

where $r=\mbox{gcd}(n_{\chi}T_{\chi},n_{\psi}T_{\psi})$, and

$\displaystyle N_{\chi}\equiv n_{\chi}T_{\chi}\,,\quad N_{\psi}\equiv
n_{\psi}T_{\psi}\,.$ (8)

Yet, we can find values of $\alpha_{2}$ and $\beta_{2}$ that leave the
discrete subgroup $\mathbb{Z}_{p_{\psi}N_{\psi}+p_{\chi}N_{\chi}}\subset
U(1)_{2}$ invariant in the color background, where $p_{\psi}$ and $p_{\chi}$
are arbitrary integers. In Appendix A, we show that most of the
$\mathbb{Z}_{p_{\psi}N_{\psi}+p_{\chi}N_{\chi}}$ elements belong to $U(1)_{A}$
and that only a subgroup
$\mathbb{Z}_{r}\subset\mathbb{Z}_{p_{\psi}N_{\psi}+p_{\chi}N_{\chi}}$, which
is $p_{\chi}$ and $p_{\psi}$-independent, can potentially act as a genuine
symmetry on the fermions. Also, we can always choose $\mathbb{Z}_{r}$ to act
solely on $\chi$:

$\displaystyle\mathbb{Z}_{r}:\quad(\psi,\chi)\longrightarrow\left(\psi,e^{i\frac{2\pi\ell}{r}}\chi\right)\,,\quad\ell=0,1,2,...,r-1\,.$
(9)

Yet, one must check that $\mathbb{Z}_{r}$ or a subgroup of it cannot be
absorbed in the centers of the color or flavor groups, which leaves a proper
subgroup of $\mathbb{Z}_{r}$ as the genuine discrete symmetry. This will be
checked on a case-by-case basis. In the following, we will use
$\mathbb{Z}_{p}^{d\chi}\subseteq\mathbb{Z}_{r}$ to denote the genuine discrete
chiral symmetry. For completeness, we remind the reader that the fermion
number symmetry $\mathbb{Z}_{2}^{F}=(-1)^{F}$ operates on $\psi$ and $\chi$ as
$(\psi,\chi)\longrightarrow-(\psi,\chi)$.

Finally, we also note that when $N$ is even, the theory is endowed with a
$\mathbb{Z}^{(1)}_{2}$ $1$-form center symmetry acting on the fundamental
Wilson loops. In summary, the good global symmetry of the theory is

$\displaystyle G^{\mbox{g}}\sim SU(n_{\psi})\times SU(n_{\chi})\times
U(1)_{A}\times\mathbb{Z}_{p}^{d\chi}\times\mathbb{Z}_{\scriptsize\mbox{gcd}(N,2)}^{(1)}\,,$
(10)

where the tilde indicates that this is the correct group modulo a discrete
group needed to fix the faithful global symmetry. Thus, the faithful global
symmetry is a quotient group.

To determine the correct quotient group, we follow Anber:2019nze ;
Anber:2021iip . Here, we keep our treatment short as the details can be found
in Anber:2021iip . We put the theory on a general compact $4$-D spin manifold
$\mathbb{M}^{4}$, define a principal bundle of the continuous part of the
global symmetry $G^{\scriptsize\mbox{g}}$ on $\mathbb{M}^{4}$, and take the
transition functions of $G^{\scriptsize\mbox{g}}$ to act on fibers by left
multiplication. Spinors are sections of the bundle, and we use the notations
$\psi_{i}$ and $\chi_{i}$ for their values on a local patch
$U_{i}\subset\mathbb{M}^{4}$. We denote the transition functions of the color
$SU(N)$, (non-abelian) flavor, and $U(1)_{A}$ group as $g$, $f$, and $u$,
respectively, along with the proper superscript to distinguish those of $\psi$
and $\chi$. On the overlap $U_{i}\cap U_{j}$ we have

$\displaystyle\psi_{i}=(g^{\psi},f^{\psi},u^{\psi})_{ij}\,\psi_{j}~{},\quad\chi_{i}=(g^{\chi},f^{\chi},u^{\chi})_{ij}\,\chi_{j}~{}.$
(11)

The fermions are well defined on $\mathbb{M}^{4}$ provided they satisfy the
cocycle condition (a necessary consistency condition) on the triple overlap
$U_{i}\cap U_{j}\cap U_{k}$. Now, we turn on background gauge fields for
centers of the gauge, flavor, and $U(1)_{A}$ groups and determine the most
general combination compatible with the cocycle condition 111See Anber:2020qzb
; Anber:2023urd ; Anber:2020xfk ; Nakajima:2022jxg  for applications of the
anomalies resulting from turning on these fluxes., which reads

$\displaystyle\left(g^{\psi},f^{\psi},u^{\psi}\right)_{ij}\circ\left(g^{\psi},f^{\psi},u^{\psi}\right)_{jk}\circ\left(g^{\psi},f^{\psi},u^{\psi}\right)_{ki}=\left(z_{c},z_{f},z_{u}\right)\quad{\rm
with}\quad z_{c}z_{f}z_{u}=1~{},$ (12)

where $z$’s refer to the center elements: $z_{c}\in\mathbb{Z}_{N/{\rm
gcd}(N,2)}$, $z_{f}\in\mathbb{Z}_{n_{\psi}}$, and $z_{u}\in U(1)_{A}$. The
condition $z_{c}z_{f}z_{u}=1$ is required for the equivalence relation

$\displaystyle\left(z_{c},z_{f},z_{u}\right)\sim(1,1,1)~{},$ (13)

which is needed to obtain the correct compatibility condition. Similar
expressions hold for the cocycle condition of $\chi$. The following two
equations give the consistency (compatibility) conditions

$\displaystyle\psi$ $\displaystyle:$
$\displaystyle\quad\underbrace{e^{i2\pi\frac{2m}{N}}}_{z_{c}}\underbrace{e^{i2\pi\frac{pk}{N-4}}}_{z_{\psi}}\underbrace{e^{-i2\pi
s\frac{(N+4)(N-2)}{kr}}}_{z_{u}}=1\,,$ $\displaystyle\chi$ $\displaystyle:$
$\displaystyle\quad\underbrace{e^{-i2\pi\frac{2m}{N}}}_{z_{c}}\underbrace{e^{-i2\pi\frac{p^{\prime}k}{N+4}}}_{z_{\chi}}\underbrace{e^{i2\pi
s\frac{(N-4)(N+2)}{kr}}}_{z_{u}}=1\,.$ (14)

Here, $m\in\mathbb{Z}_{\scriptsize N/\mbox{gcd}(N,2)}$,
$p\in\mathbb{Z}_{n_{\psi}}$, $p^{\prime}\in\mathbb{Z}_{n_{\chi}}$ and $s$ is a
$U(1)_{A}$ parameter. The factor of $2$ that appears in $z_{c}$ accounts for
the $N$-ality of $\psi$ and $\chi$, and the negative sign that appears in the
$z_{c}$ factor in the second line accounts for the fact that $\chi$ transforms
in the complex conjugate of the $2$-index anti-symmetric representation. We
also take $\psi$ to transform in the fundamental representation of
$SU(n_{\psi})$ and $\chi$ to transform in the anti-fundamental representation
of $SU(n_{\chi})$. Following Anber:2021iip , we shall dub the discrete color-
flavor-$U(1)_{A}$ fluxes as the CFU fluxes. The full set of solutions of (14)
determines the quotient group in (10). These solutions will be found on a
case-by-case basis. In general, we divide (10) by $\mathbb{Z}_{\scriptsize
N/\mbox{gcd}(N,2)}\times\mathbb{Z}_{(N-4)/k}\times\mathbb{Z}_{(N+4)/k}$ or a
subgroup of it.

Once a non-trivial solution of (14) is found, we can calculate the topological
charges associated with the center fluxes, which are fractional charges in
general. Let $\mathbb{M}^{4}$ admit two independent $2$-cycles and let two
integers, e.g., $m_{1}$ and $m_{2}$, account for the number of quanta piercing
through them. For example, we can take $\mathbb{M}^{4}=\mathbb{T}^{4}$, a
$4$-torus with a period length $L$, and turn on the gauge fields that are
compatible with the cocycle condition:

$\displaystyle a_{1}^{c}$ $\displaystyle=$ $\displaystyle\frac{2\pi
m_{1}}{L^{2}}\bm{H}_{c}\cdot\bm{\nu}_{c}\,,\quad a_{2}^{c}=0\,,\quad
a_{3}^{c}=\frac{2\pi m_{2}}{L^{2}}\bm{H}_{c}\cdot\bm{\nu}_{c}\,,\quad
a_{4}^{c}=0\,,$ $\displaystyle A_{1}^{\psi}$ $\displaystyle=$
$\displaystyle\frac{2\pi
p_{1}}{L^{2}}\bm{H}_{\psi}\cdot\bm{\nu}_{\psi}\,,\quad A_{2}^{\psi}=0\,,\quad
A_{3}^{\psi}=\frac{2\pi p_{2}}{L^{2}}\bm{H}_{\psi}\cdot\bm{\nu}_{\psi}\,,\quad
A_{4}^{\psi}=0\,,$ $\displaystyle A_{1}^{\chi}$ $\displaystyle=$
$\displaystyle\frac{2\pi
p_{1}^{\prime}}{L^{2}}\bm{H}_{\chi}\cdot\bm{\nu}_{\chi}\,,\quad
A_{2}^{\chi}=0\,,\quad A_{3}^{\chi}=\frac{2\pi
p_{2}^{\prime}}{L^{2}}\bm{H}_{\chi}\cdot\bm{\nu}_{\chi}\,,\quad
A_{4}^{\chi}=0\,,$ $\displaystyle A_{1}^{u}$ $\displaystyle=$
$\displaystyle\frac{2\pi s_{1}}{L^{2}}\,,\quad A_{2}^{u}=0\,,\quad
A_{3}^{u}=\frac{2\pi s_{2}}{L^{2}}\,,\quad A_{4}^{u}=0\,.$ (15)

$a_{\mu}^{c}$, $A_{\mu}^{\psi}$, $A_{\mu}^{\chi}$, and $A^{u}_{\mu}$ are the
background gauge fields of the center of the color, $SU(n_{\psi})$ flavor,
$SU(n_{\chi})$, flavor, and $U(1)_{A}$, respectively. The bold-face letters
$\bm{H}\equiv(H_{1},...,H_{N-1})$ are the Cartan generators of $SU(N)$ group,
while $\bm{\nu}\equiv(\nu_{1},...,\nu_{N-1})$ is a weight in the defining
representation of the group. Notice that the integers
$m_{1,2},p_{1,2},p^{\prime}_{1,2},s_{1,2}$ are the same integers that solve
the consistency conditions (14). Given the set of the background fields (15),
one immediately obtains the topological charges defined as

$\displaystyle Q_{c}=\int_{\mathbb{T}^{4}}\frac{\mbox{tr}\left[f^{c}\wedge
f^{c}\right]}{8\pi^{2}}\,,Q_{\psi}=\int_{\mathbb{T}^{4}}\frac{\mbox{tr}\left[F^{\psi}\wedge
F^{\psi}\right]}{8\pi^{2}}\,,Q_{\chi}=\int_{\mathbb{T}^{4}}\frac{\mbox{tr}\left[F^{\chi}\wedge
F^{\chi}\right]}{8\pi^{2}}\,,Q_{u}=\int_{\mathbb{T}^{4}}\frac{F^{u}\wedge
F^{u}}{8\pi^{2}}\,,$ (16)

and $f^{c}$, $F^{\psi,\chi,u}$ are the field strengths of the corresponding
background. Substituting (15) into (16), we obtain

$\displaystyle Q_{c}$ $\displaystyle=$ $\displaystyle
k_{c}-\frac{m_{1}m_{2}}{N}\,,\quad
Q_{\psi}=k_{\psi}-\frac{p_{1}p_{2}k}{N-4}\,,$ $\displaystyle\quad Q_{\chi}$
$\displaystyle=$ $\displaystyle
k_{\chi}-\frac{p^{\prime}_{1}p^{\prime}_{2}k}{N+4}\,,\quad
Q_{u}=(s_{1}-k_{1})(s_{2}-k_{2})\,,$ (17)

and $k_{c},k_{\psi},k_{\chi},k_{1},k_{2}\in\mathbb{Z}$ are arbitrary integers
that are always allowed. These fluxes will support fermion zero modes, and the
Dirac indices give their number:

$\displaystyle{\cal I}_{\psi}$ $\displaystyle=$ $\displaystyle
n_{\psi}T_{\psi}Q_{c}+\mbox{dim}_{\psi}Q_{\psi}+\mbox{dim}_{\psi}n_{\psi}q_{\psi}^{2}Q_{u}\,,$
$\displaystyle{\cal I}_{\chi}$ $\displaystyle=$ $\displaystyle
n_{\chi}T_{\chi}Q_{c}+\mbox{dim}_{\chi}Q_{\chi}+\mbox{dim}_{\chi}n_{\chi}q_{\chi}^{2}Q_{u}\,,$
(18)

and $\mbox{dim}_{\psi}=\frac{N(N+1)}{2}$, $\mbox{dim}_{\chi}=\frac{N(N-1)}{2}$
are the dimensions of the representations. Dirac indices count the number of
the Weyl zero modes in the background of center fluxes. The integrality of the
indices can work as a check on the consistency of the fluxes on
$\mathbb{M}^{4}$.

One may also turn on the CFU fluxes on nonspin $\mathbb{M}^{4}$. A nonspin
manifold does not admit fermions in the sense that there is an obstruction in
lifting the $SO(4)$ rotation group bundle to a $\mbox{Spin}(4)$ bundle on
$\mathbb{M}^{4}$. A diagnosis of a non-spin manifold is that the Dirac index
of a Weyl fermion, ${\cal
I}=\frac{1}{196\pi^{2}}\int_{\mathbb{M}^{4}}\mbox{tr}R\wedge R$, where $R$ is
the curvature $2$-form, is non-integer. An example of a nonspin manifold is
$\mathbb{CP}^{2}$, which has
$\frac{1}{196\pi^{2}}\int_{\mathbb{CP}^{2}}\mbox{tr}R\wedge R=-\frac{1}{8}$ .
To put a Weyl fermion on $\mathbb{CP}^{2}$, we need to excite a $U(1)$ flux
$F$ through its $2$-cycle $\mathbb{CP}^{1}\subset\mathbb{CP}^{2}$ and demand
that $\int_{\mathbb{CP}^{1}}F\in\pi(2\mathbb{Z}+1)$, which implies
$\frac{1}{8\pi^{2}}\int_{\mathbb{CP}^{2}}F\wedge F\in\frac{\mathbb{Z}}{8}$.
Now, one can easily check the integrality of the Dirac index
$\frac{1}{196\pi^{2}}\int_{\mathbb{CP}^{2}}\mbox{tr}R\wedge
R+\frac{1}{8\pi^{2}}\int_{\mathbb{CP}^{2}}F\wedge F\in\mathbb{Z}$, and thus,
the fermions are well-defined on $\mathbb{CP}^{2}$ in the presence of such
$U(1)$ fluxes. Here, although one cannot define a $\mbox{Spin}(4)$ bundle on
pure $\mathbb{CP}^{2}$, in the sense that the corresponding cocycle condition
fails on a triple overlap, nonetheless, we can define the $\mbox{Spin}_{c}(4)$
structure $\mbox{Spin}(4)\times U(1)/\mathbb{Z}_{2}$ in the presence of the
$U(1)$ background.

This idea can be generalized in the presence of the CFU fluxes; see
Anber:2020gig  for details. One just needs to replace the consistency
conditions (14) with

$\displaystyle\psi$ $\displaystyle:$
$\displaystyle\quad\underbrace{e^{i2\pi\frac{2m}{N}}}_{z_{c}}\underbrace{e^{i2\pi\frac{pk}{N-4}}}_{z_{\psi}}\underbrace{e^{-i2\pi
s\frac{(N+4)(N-2)}{kr}}}_{z_{u}}=-1\,,$ $\displaystyle\chi$ $\displaystyle:$
$\displaystyle\quad\underbrace{e^{-i2\pi\frac{2m}{N}}}_{z_{c}}\underbrace{e^{-i2\pi\frac{p^{\prime}k}{N+4}}}_{z_{\chi}}\underbrace{e^{i2\pi
s\frac{(N-4)(N+2)}{kr}}}_{z_{u}}=-1\,.$ (19)

The minus sign on the right-hand side compensates for the minus sign arising
from parallel transporting the spinor fields around appropriate closed paths
in $\mathbb{CP}^{2}$; see the detailed discussion in Anber:2020gig . Given
that a solution, $m\in\mathbb{Z}_{\scriptsize N/\mbox{gcd}(N,2)}$,
$p\in\mathbb{Z}_{n_{\psi}}$, $p^{\prime}\in\mathbb{Z}_{n_{\chi}}$ and $s$, to
(19) can be found, the topological charges corresponding to the CFU fluxes and
gravity are given by (see Anber:2020gig )

$\displaystyle Q_{c}$
$\displaystyle=\frac{m^{2}}{2}\left(1-\frac{1}{N}\right)~{},\quad
Q_{\psi}=\frac{p^{2}}{2}\left(1-\frac{k}{N-4}\right)~{},$ $\displaystyle
Q_{\chi}$ $\displaystyle=\frac{p^{\prime
2}}{2}\left(1-\frac{k}{N+4}\right)~{},\quad Q_{u}=\frac{1}{2}s^{2}\,,\quad
Q_{g}=-\frac{1}{8}~{}\,.$ (20)

The Dirac-indices of $\psi$ and $\chi$ are

$\displaystyle{\cal I}_{\psi}^{\mathbb{CP}^{2}}$ $\displaystyle=$
$\displaystyle
n_{\psi}T_{\psi}Q_{c}+\mbox{dim}_{\psi}Q_{\psi}+\mbox{dim}_{\psi}n_{\psi}\left(q_{\psi}^{2}Q_{u}+Q_{g}\right)~{},$
$\displaystyle{\cal I}_{\chi}^{\mathbb{CP}^{2}}$ $\displaystyle=$
$\displaystyle
n_{\chi}T_{\chi}Q_{c}+\mbox{dim}_{\chi}Q_{\chi}+\mbox{dim}_{\chi}n_{\chi}\left(q_{\chi}^{2}Q_{u}+Q_{g}\right)~{},$
(21)

which are always integers. Except for $(N=6,k=2)$ and $(N=10,k=2)$ in Table 1,
we can always find solutions to (19), and thus, we can put these theories on
$\mathbb{CP}^{2}$.

###  2.2 Anomalies

The theory has a set of ’t Hooft anomalies that can help constrain the
possible IR phases. In the following, we list the ’t Hooft anomalies we shall
encounter in our study.

###  (I) $\left[SU(n_{\psi})\right]^{3}$ and $\left[SU(n_{\chi})\right]^{3}$
anomalies

These are perturbative (triangle) anomalies and their inflow from $5$-D to
$4$-D is captured via $5$-D Chern-Simons theories:

$\displaystyle\left[SU(n_{\psi})\right]^{3}$ $\displaystyle:$
$\displaystyle\quad\exp\left[i\,\mbox{dim}_{\psi}\int_{{\mathbb{M}}^{5}}\omega_{5}(A^{\psi})\right]\,,$
$\displaystyle\left[SU(n_{\chi})\right]^{3}$ $\displaystyle:$
$\displaystyle\quad\exp\left[i\,\mbox{dim}_{\chi}\int_{{\mathbb{M}}^{5}}\omega_{5}(A^{\chi})\right]\,,$
(22)

where $A^{\psi}$ and $A^{\chi}$ are the $SU(n_{\psi})$ and $SU(n_{\chi})$
$1$-form background gauge fields, extended from $4$-D to $5$-D, and
$\omega_{5}(A)$ is the $5$-D Chern-Simons form defined via the descent
equation:

$\displaystyle d\omega_{5}(A)=\frac{1}{3!(2\pi)^{2}}\mbox{tr}_{\Box}F^{3}\,,$
(23)

and $F$ is the $2$-form field strength of $A$.

### (II) $U(1)_{A}$\- and $\mathbb{Z}_{p}^{d\chi}$-gravitational anomalies

These anomalies are captured via the $5$-D anomaly inflow actions:

$\displaystyle U(1)_{A}[\mbox{grav}]$ $\displaystyle:$
$\displaystyle\quad\exp\left[i\left(q_{\psi}n_{\psi}\mbox{dim}_{\psi}+q_{\chi}n_{\chi}\mbox{dim}_{\chi}\right)\int_{\mathbb{M}^{5}}A^{u}\wedge\frac{p_{1}(\mathbb{M}^{5})}{24}\right]\,,$
$\displaystyle\mathbb{Z}_{p}^{d\chi}[\mbox{grav}]$ $\displaystyle:$
$\displaystyle\quad\exp\left[i\left(n_{\chi}\mbox{dim}_{\chi}\right)\int_{\mathbb{M}^{5}}A^{d\chi}\wedge\frac{p_{1}(\mathbb{M}^{5})}{24}\right]\,.$
(24)

The $1$-form gauge fields $A^{u}$ and $A^{d\chi}$ are the backgrounds of
$U(1)_{A}$ and $\mathbb{Z}_{p}^{d\chi}$, respectively.
$p_{1}(\mathbb{M}^{5})=-\frac{1}{8\pi^{2}}R\wedge R$ is the first Pontryagin
number and $R$ is the curvature $2$-form. On a spin manifold, we have
$\int_{\mathbb{M}^{4}}p_{1}(\mathbb{M}^{4})\in 48\,\mathbb{Z}$, and thus,
there are $2$ zero modes per Weyl fermion in a gravitational background. Under
$U(1)_{A}$ and $\mathbb{Z}_{p}^{d\chi}$ transformations we have
$A^{u}\longrightarrow A^{u}+d\lambda^{u}$ with $\oint
d\lambda^{u}=2\pi\mathbb{Z}$ and $A^{d\chi}\longrightarrow
A^{d\chi}+d\lambda^{d\chi}$, with $\oint
d\lambda^{d\chi}=\frac{2\pi\mathbb{Z}}{p}$, and the anomaly inflow actions
produce the $4$-D anomalies.

The result (24) is “perturbative” as it can be seen from a triangle diagram
with two vertices that couple the fermions to a gravitational background via
the energy-momentum tensor, while the third vertex couples the fermions to an
external $U(1)_{A}$ or $\mathbb{Z}_{p}^{d\chi}$ sources.

### (III) CFU anomalies

These anomalies were identified in Anber:2019nze ; however, see
Tanizaki:2018wtg ; Shimizu:2017asf  for earlier encounters. They are anomalies
of $U(1)_{A}$ and $\mathbb{Z}_{p}^{d\chi}$ symmetries in the background of the
CFU fluxes that are supported on a general spin manifold. As was shown in
Anber:2021iip , $5$-D anomaly inflow actions can also capture them. However,
we find it more convenient to express such anomalies in terms of the non-
trivial phases that are acquired by the partition function ${\cal Z}$ under
the action of $U(1)_{A}$ and $\mathbb{Z}_{p}^{d\chi}$ symmetries in the
background of the CFU fluxes:

$\displaystyle U(1)_{A}[\mbox{CFU}]$ $\displaystyle:$ $\displaystyle{\cal
Z}\longrightarrow e^{i2\pi\alpha\left(q_{\psi}{\cal I}_{\psi}+q_{\chi}{\cal
I}_{\chi}\right)}{\cal Z}\,,$
$\displaystyle\mathbb{Z}_{p}^{d\chi}[\mbox{CFU}]$ $\displaystyle:$
$\displaystyle{\cal Z}\longrightarrow e^{i\frac{2\pi}{p}{\cal I}_{\chi}}{\cal
Z}\,,$ (25)

and the Dirac indices ${\cal I}_{\psi}$ and ${\cal I}_{\chi}$ are given in
(18). The contribution from the color topological charge $Q_{c}$ drops out in
the computation of the $U(1)_{A}[\mbox{CFU}]$ anomaly, as can be easily
checked, since $U(1)_{A}$ is a good symmetry in the background of the color
flux. This is not the case with $\mathbb{Z}_{p}^{d\chi}[\mbox{CFU}]$ anomaly,
where $Q_{c}$ contributes to the anomaly. As we shall discuss, this
observation has important consequences for anomaly matching in the IR.

It is also important to notice that the perturbative anomalies
$U(1)_{A}[SU(n_{\psi})]^{2}$, $U(1)_{A}[SU(n_{\chi})]^{2}$,
$\mathbb{Z}_{p}^{d\chi}[SU(n_{\chi})]^{2}$, and $[U(1)_{A}]^{3}$ are a subset
of the CFU anomalies, obtained by turning off the center fluxes and keeping
only the integer topological charges in (17). Again, one can express them
using anomaly inflow actions as

$\displaystyle U(1)_{A}[SU(n_{\psi})]^{2}$ $\displaystyle:$
$\displaystyle\quad\exp\left[iq_{\psi}n_{\psi}\mbox{dim}_{\psi}\int_{\mathbb{M}^{5}}A^{u}\wedge\frac{\mbox{tr}\left[F^{\psi}\wedge
F^{\psi}\right]}{8\pi^{2}}\right]\,,$ $\displaystyle
U(1)_{A}[SU(n_{\chi})]^{2}$ $\displaystyle:$
$\displaystyle\quad\exp\left[iq_{\chi}n_{\chi}\mbox{dim}_{\chi}\int_{\mathbb{M}^{5}}A^{u}\wedge\frac{\mbox{tr}\left[F^{\chi}\wedge
F^{\chi}\right]}{8\pi^{2}}\right]\,,$
$\displaystyle\mathbb{Z}_{p}^{d\chi}[SU(n_{\chi})]^{2}$ $\displaystyle:$
$\displaystyle\quad\exp\left[in_{\chi}\mbox{dim}_{\chi}\int_{\mathbb{M}^{5}}A^{d\chi}\wedge\frac{\mbox{tr}\left[F^{\chi}\wedge
F^{\chi}\right]}{8\pi^{2}}\right]\,,$ $\displaystyle\left[U(1)_{A}\right]^{3}$
$\displaystyle:$
$\displaystyle\quad\exp\left[i\left(q_{\psi}^{3}n_{\psi}\mbox{dim}_{\psi}+q_{\chi}^{3}n_{\chi}\mbox{dim}_{\chi}\right)\int_{\mathbb{M}^{5}}A^{u}\wedge\frac{F^{u}\wedge
F^{u}}{24\pi^{2}}\right]\,.$ (26)

In addition, for $N$ even, the CFU anomalies encompass the $U(1)_{A}$
$0$-form/ $\mathbb{Z}_{2}^{[1]}$ $1$-form as well as the
$\mathbb{Z}_{r}^{d\chi}$ $0$-form/ $\mathbb{Z}_{2}^{[1]}$ $1$-form mixed
anomalies. These can be easily found by turning off the flavor and the
$U(1)_{A}$ fluxes. In practice, one uses (25), (17), and (18), after setting
$p_{1,2}=p^{\prime}_{1,2}=s_{1,2}=0$ and $m_{1}=m_{2}=N/2$. This choice
enforces the consistency conditions (14) and gives $Q_{\psi}=Q_{\chi}=Q_{u}=0$
and $Q_{c}=\frac{N}{4}$.

One may also use the CFU fluxes on $\mathbb{CP}^{2}$ to calculate the
$U(1)_{A}\left[\mbox{CFU}\right]$ and $\mathbb{Z}_{p}\left[\mbox{CFU}\right]$
anomalies, which sometimes are more restrictive than the corresponding
anomalies on a spin manifold. We use the Dirac indices on ${\mathbb{CP}^{2}}$,
as given by (21), to find

$\displaystyle U(1)_{A}\left[\mbox{CFU}\right]_{\mathbb{CP}^{2}}$
$\displaystyle:$ $\displaystyle{\cal Z}\longrightarrow
e^{i2\pi\alpha\left[q_{\psi}{\cal I}_{\psi}^{\mathbb{CP}^{2}}+q_{\chi}{\cal
I}_{\chi}^{\mathbb{CP}^{2}}\right]}{\cal Z}~{},$
$\displaystyle\mathbb{Z}_{p}^{d\chi}\left[\mbox{CFU}\right]_{\mathbb{CP}^{2}}$
$\displaystyle:$ $\displaystyle{\cal Z}\longrightarrow e^{i\frac{2\pi}{p}{\cal
I}_{\chi}^{\mathbb{CP}^{2}}}{\cal Z}~{}.$ (27)

From here on, we shall write all anomalies in terms of their phases to reduce
clutter. For example, instead of (25), we write:

$\displaystyle U(1)_{A}[\mbox{CFU}]:q_{\psi}{\cal I}_{\psi}+q_{\chi}{\cal
I}_{\chi}\,,\quad\mathbb{Z}_{p}^{d\chi}[\mbox{CFU}]:{\cal I}_{\chi}\,.$ (28)

### (IV) Anomalies of discrete groups

Here, we consider anomalies of a discrete symmetry $\mathbb{Z}_{n}$, where $n$
is a general positive integer. An example of the discrete symmetry is the
$\mathbb{Z}_{p}^{d\chi}$ chiral symmetry or a discrete subgroup of $U(1)_{A}$
left unbroken in the IR. The proper way to detect anomalies of discrete
symmetries is to use the Dai-Freed prescription Dai:1994kq ; Witten:2019bou .
The idea stems from the fact that a chiral massless fermion defined on
$\mathbb{M}^{4}$ can be realized as the chiral zero mode residing on the
boundary $\mathbb{M}^{4}$ of a $5$-dimensional manifold $\mathbb{M}^{5}$ that
is endowed with massive fermions, with $\mathbb{Z}_{n}$ turned on in the
$5$-dimensional manifold. One can also consider a different $5$-dimensional
manifold $\mathbb{M}^{\prime 5}$ with the same boundary $\mathbb{M}^{4}$. If
the partition functions defined on $\mathbb{M}^{5}$ and $\mathbb{M}^{\prime
5}$ have the same phase, then the theory on $\mathbb{M}^{4}$ is uniquely
defined and anomaly-free; otherwise, it is anomalous. Applying the Dai-Freed
prescription to study the IR phases of strongly-coupled theories is
innovative. However, see Razamat:2020kyf ; Anber:2023urd  for previous
applications222Also, see Garcia-Etxebarria:2018ajm ; Davighi:2019rcd  for
applications of Dai-Freed anomalies in particle physics..

If $\mathbb{M}^{4}$ is a spin manifold, the geometrical obstruction of
uniquely extending a $4$-D theory to a $5$-D bulk can be inferred by computing
the bordism group $\Omega_{5}^{\scriptsize\mbox{Spin}}(B\mathbb{Z}_{n})$,
where $B\mathbb{Z}_{n}$ is the classifying space of $\mathbb{Z}_{n}$333A
classifying space of symmetry $G$ is an infinite dimensional space with the
property that any principal $G$-bundle on a manifold $\mathbb{M}$ is the
pullback via some map $f:\mathbb{M}\longrightarrow BG$. Then, the set of
topologically distinct principal $G$-bundles over $\mathbb{M}$ is equivalent
to the set of the homotopy classes of maps from $\mathbb{M}$ to $BG$.. If
$\Omega_{5}^{\scriptsize\mbox{Spin}}(B\mathbb{Z}_{n})$ is non-trivial, the
theory might have a nonperturbative anomaly. To find the anomaly, one computes
the $\eta$-invariant, a resolvent of the spectral asymmetry of the Dirac
operator, on specific closed $5$-dimensional spin manifolds that can detect
the anomaly. For example, one puts the theory on Lens spaces to gauge
$\mathbb{Z}_{n}$ and discover whether the theory exhibits a nonperturbative
anomaly. For $n$ even, $n=2m$, one can take $\mathbb{M}^{4}$ to be nonspin by
employing the twisted symmetry group
$\mbox{Spin}^{\mathbb{Z}_{2m}}(4)=(\mbox{Spin}(4)\times\mathbb{Z}_{2m})/\mathbb{Z}_{2}$
instead of $\mbox{Spin}(4)$. Here, one needs to compute the $\eta$-invariant
that detects the bordism group
$\Omega_{5}^{\scriptsize\mbox{Spin}^{\mathbb{Z}_{2m}}}$.

The computations of the relevant $\eta$-invariants were carried out in
Hsieh:2018ifc  (see also Davighi:2020uab  for an alternative perspective). For
a theory of a left-handed Weyl fermion with a charge $s$ under
$\mathbb{Z}_{n}$ defined on a spin manifold, the anomaly is given by the pair
of phases:

$\displaystyle\left\\{(n^{2}+3n+2)s^{3}\,\mbox{mod}\,6n\,,2s\,\mbox{mod}\,n\right\\}\,.$
(29)

This pair can be thought of as contributions from $[\mathbb{Z}_{n}]^{3}$ and
mixed $\mathbb{Z}_{n}\left[\mbox{grav}\right]$ anomalies. Indeed, the second
entry in (29) is precisely the anomaly we computed in the second line of (24).
The first entry can be obtained from pure $[U(1)]^{3}$ anomaly by restricting
$U(1)$ to a $\mathbb{Z}_{n}$ discrete group; this is the Ibanez-Ross anomaly
we comment on below. Also, a Weyl fermion defined on a twisted background and
carrying a charge $s$ under $\mathbb{Z}_{2m}$ has an anomaly given by the
pair444More generally, the bordism group
$\Omega_{5}^{\scriptsize\mbox{Spin}^{\mathbb{Z}_{2m}}}\cong\mathbb{Z}_{a}\times\mathbb{Z}_{b}$,
where $a,b$, and the associated anomalies are given by Eqs. (2.11)- (2.13) in
Anber:2023urd . In the present work, Eq. (30) suffices to tackle the theories
at hand.

$\displaystyle\left\\{\left((2m^{2}+m+1)s^{3}-(m+3)s\right)\,\mbox{mod}\,48m\,,\left(ms^{3}+s\right)\,\mbox{mod}\,2m\right\\}\,.$
(30)

The charge $s$ is assumed to be odd such that the fermion transforms under the
$\mathbb{Z}_{2}$ subgroup of $\mathbb{Z}_{2m}$. Generally, the anomaly (30) is
more restrictive than (29). We shall use both (30) and (29) to constrain our
theories. It is important to note that $\mathbb{Z}_{2}$ symmetry is anomaly-
free, as can be easily seen from (29) and (30). This observation will play an
essential role in the IR anomaly matching by condensates, as many of them
break the global symmetries to $\mathbb{Z}_{2}$, as we discuss below.

We also comment on the Ibanez-Ross anomaly-matching conditions Ibanez:1991hv .
These are obtained from $[U(1)]^{3}$ and $U(1)[\mbox{grav}]$ anomalies by
restricting $U(1)$ to a $\mathbb{Z}_{n}$ subgroup. The Ibanez-Ross anomaly-
cancellation conditions read:

$\displaystyle s^{3}=p^{\prime}n+\frac{r^{\prime}n^{3}}{8}\,,\quad
s=p^{\prime\prime}n+r^{\prime\prime}n\,,$ (31)

where $p^{\prime},r^{\prime},p^{\prime\prime},r^{\prime\prime}\in\mathbb{Z}$,
$p^{\prime}\in 3\mathbb{Z}$ if $n\in 3\mathbb{Z}$, and
$r^{\prime},r^{\prime\prime}=0$ if $n$ is odd. It can be shown that (31) and
(29) are equivalent Hsieh:2018ifc . Hence, in what follows, we use either (29)
or (30) to calculate discrete anomalies.

Finally, we also may have a discrete anomaly of the form
$\mathbb{Z}_{m}[\mathbb{Z}_{n}]^{2}$. Such an anomaly can descend from
$U(1)_{A}[\mbox{CFU}]$ anomaly after a given condensate breaks $U(1)_{A}$ down
to a discrete subgroup. Let $s$ and $s^{\prime}$ be the charges of a left-
handed Weyl fermion under $\mathbb{Z}_{n}$ and $\mathbb{Z}_{m}$, respectively.
Then, the anomaly cancelation condition, which follows from the Ibanez-Ross
conditions, is given by Csaki:1997aw

$\displaystyle
s^{2}s^{\prime}=p^{\prime}\,\mbox{gcd}(m,n)+\frac{p^{\prime\prime}}{8}mn^{2}\,,$
(32)

where $p^{\prime},p^{\prime\prime}\in\mathbb{Z}$ and $p^{\prime\prime}$ can be
non-vanishing only if $n$ and $m$ are even. Notice that this anomaly is
trivial when $\mathbb{Z}_{m}=\mathbb{Z}_{2}$.

##  3 Anomaly matching and the IR phase

###  3.1 IR anomaly matching

’t Hooft anomalies preclude a trivially gapped IR phase: a theory with ’t
Hooft anomaly must have gapless excitations, degenerate vacua, or symmetry-
preserving topological quantum field theory (TQFT). In this work, we use the
safe assumption that the gauge group does not break spontaneously under its
strong dynamics. The breaking of a gauge group is under control in the
presence of scalars at weak coupling, an ingredient absent from our theory
from the get-go555Tumbling, Raby:1979my , is a mechanism by which the
breakdown of a gauge group occurs without the aid of fundamental scalar
fields. We do not discuss tumbling in this work.. This leaves us with three
possible IR scenarios.

### (I) Conformal fixed point

In the first scenario, the theory flows to a conformal field theory (CFT).
When the CFT is weakly coupled, the renormalization group flow from the UV to
the IR is very slow. The UV matter content (fermions) can be considered the IR
gapless excitations, and the anomalies are automatically matched. Anomaly
matching by strongly interacting CFT is still an open problem; we have nothing
to say here. The existence of a well-controlled Banks-Zaks fixed point implies
a strictly weakly coupled CFT. However, such a reliable fixed point can only
be obtained in the large-$N$ limit. The $3$-loop $\beta$-function reads

$\displaystyle\beta(g)=$
$\displaystyle-\beta_{0}\frac{g^{3}}{(4\pi)^{2}}-\beta_{1}\frac{g^{5}}{(4\pi)^{4}}-\beta_{2}\frac{g^{7}}{(4\pi)^{6}}\,.$
(33)

If $\beta_{0}>0$ and $\beta_{1}<0$, the theory flows to an IR fixed point,
$g^{2}_{*}=-\frac{(4\pi)^{2}\beta_{0}}{\beta_{1}}$, up to corrections from
$\beta_{2}$. In the large-$N$ limit and close to the boundary of the
asymptotic-freedom region, the contribution from the third term is suppressed
compared to the second term, and thus, this term and higher-order terms can be
safely neglected. Our theories, however, do not admit large-$N$ analysis. Yet,
as we shall discuss, in a few cases, the numerical value of the third term is
extremely small compared to the first two terms, so one can conclude that such
a weakly-coupled CFT exists. In Appendix B, we work out the fixed points using
the $2$ and $3$-loop $\beta$-functions. We consider values of
$\frac{g_{*}^{2}}{4\pi}<0.1$, using both the $2$-loop and $3$-loop
calculations, small enough to conclude that the theory has an IR fixed point.
Also, the perturbative nature of the $\beta$-function calculations will be
trusted when the third term in (33) is small compared to the first two terms.
More stringent coupling constant values at the fixed point could also be
assumed. This, however, will only mean doing more work to find out the fate of
the IR phases of such theories.

### (II) Composite massless fermions

In the second scenario, the theory becomes strongly coupled; it confines (for
$N$ even), preserves the global symmetries, and flows to a phase of composite
massless fermions. This can happen in the non-bosonic theories $N=5,6,10$. We
sketch how one can systematically search for such composites. Let ${\cal
F}_{i}$ be a gauge-invariant fermionic operator (a composite that transforms
as a left-handed Weyl fermion under the Lorentz group) built of $\psi$ and
$\chi$:

$\displaystyle{\cal F}_{i}=\psi^{\kappa_{{}_{i}}}\chi^{\rho_{i}}\,,$ (34)

where $\kappa_{i},\rho_{i}\in\\{0\\}\cup\mathbb{Z}^{+}$, and we suppressed the
color and spinor indices to reduce notational clutter. Insertion of gluon
fields can be used whenever fermi statistics cause ${\cal F}_{i}$ to vanish.
Using the convention that $\psi$ and $\chi$ carry $2$ and $N-2$ indices,
respectively, and demanding that ${\cal F}_{i}$ be a gauge invariant fermion
yields the two conditions:

$\displaystyle 2\kappa_{i}+(N-2)\rho_{i}\in
N\mathbb{Z}^{+}\,,\quad\kappa_{i}+\rho_{i}\in(2\mathbb{Z}^{+}-1)\,.$ (35)

The $U(1)_{A}$ charge of ${\cal F}_{i}$ is

$\displaystyle q_{{\cal
F}_{i}}=\frac{-\kappa_{i}N_{\chi}+\rho_{i}N_{\psi}}{r}\,.$ (36)

Generally, the composites ${\cal F}_{i}$ transform in higher representations
of $SU(n_{\psi})$ and $SU(n_{\chi})$, making the process of matching anomalies
containing flavor groups a daunting task. Thus, it is more convenient to start
with matching $\left[U(1)_{A}\right]^{3}$, $U(1)_{A}[\mbox{grav}]$, and
nonperturbative ${\mathbb{Z}}_{p}^{d\chi}$ anomalies. For generality, we
assume there are ${\cal N}_{i}$ copies of composites ${\cal F}_{i}$. Then,
matching these anomalies gives the conditions

$\displaystyle\sum_{i}{\cal N}_{i}q_{{\cal F}_{i}}^{3}$ $\displaystyle=$
$\displaystyle
q_{\psi}^{3}n_{\psi}\mbox{dim}_{\psi}+q_{\chi}^{3}n_{\chi}\mbox{dim}_{\chi}\,,$
$\displaystyle\sum_{i}{\cal N}_{i}q_{{\cal F}_{i}}$ $\displaystyle=$
$\displaystyle
q_{\psi}n_{\psi}\mbox{dim}_{\psi}+q_{\chi}n_{\chi}\mbox{dim}_{\chi}\,,$
$\displaystyle 2\sum_{i}{\cal N}_{i}$ $\displaystyle=$ $\displaystyle
2n_{\chi}\mbox{dim}_{\chi}\,(\mbox{mod}\,p)\,,\quad(p^{2}+3p+2)\sum_{i}{\cal
N}_{i}=(p^{2}+3p+2)n_{\chi}\mbox{dim}_{\chi}\,(\mbox{mod}\,6p)\,.$

The number of the IR fermionic species ${\cal N}=\sum_{i}{\cal N}_{i}$ is
bounded from above by the a-theorem:

$\displaystyle\underbrace{\underbrace{2(N^{2}-1)}_{\mbox{gluons}}+\frac{7}{4}\left(n_{\psi}\mbox{dim}_{\psi}+n_{\chi}\mbox{dim}_{\chi}\right)}_{\mbox{UV
degrees of freedom}}\geq\underbrace{\frac{7{\cal N}}{4}}_{\mbox{IR degrees of
freedom}}\,.$ (38)

In principle, one could systematically search for copies of composites
$\\{{\cal N}_{1},{\cal N}_{2},...\\}$ that satisfy (LABEL:the_3_conditions).
However, this would require finding the partitions of ${\cal N}$ (all integers
that their sums give ${\cal N}$), a number that grows exponentially with
$\sqrt{\cal N}$. The composites that satisfy (LABEL:the_3_conditions) must
also match the rest of the anomalies that involve the flavor groups. In all
non-bosonic theories, $N=5,6,10$, we could not find a set of composites that
matched the full set of anomalies using the systematic approach sketched
above. Simply, the algorithm takes an extremely long time, which makes such a
systematic search impractical.

In fact, we can utilize the $\mathbb{Z}_{p}^{d\chi}[\mbox{CFU}]$ anomaly to
show that in some cases, such candidates, if they exist, cannot solely match
this anomaly. This approach was used in Anber:2019nze  in the case of vector-
like theories, and we repeat it here for chiral theories. To this end, we
assume that there exists a set of gauge-invariant composite fermions that
match $[SU(n_{\psi})]^{3}$, $[SU(n_{\chi})]^{3}$, $[U(1)_{A}]^{3}$,
$\mathbb{Z}_{p}^{d\chi}[U(1)_{A}]^{2}$,
$\mathbb{Z}_{p}^{d\chi}[SU(n_{\psi})]^{2}$,
$\mathbb{Z}_{p}^{d\chi}[SU(n_{\chi})]^{2}$, $U(1)_{A}[\mbox{grav}]$, and
$\mathbb{Z}_{p}^{d\chi}[\mbox{grav}]$ anomalies. Then, we turn the CFU fluxes
on $\mathbb{M}^{4}$ and perform a $\mathbb{Z}_{p}^{d\chi}$ rotation. We denote
the UV coefficients that multiply $Q_{c}$, $Q_{\chi}$, and $Q_{u}$ in (25, 18)
by $D_{c}^{\scriptsize\mbox{UV}}\equiv n_{\chi}T_{\chi}$,
$D_{\chi}^{\scriptsize\mbox{UV}}\equiv\mbox{dim}_{\chi}$, and
$D_{u}^{\scriptsize\mbox{UV}}\equiv q_{\chi}^{2}n_{\chi}\mbox{dim}_{\chi}$.
Under a discrete chiral rotation, the UV partition function transforms as

$\displaystyle{\cal Z}_{\scriptsize\mbox{UV}}\longrightarrow
e^{i\frac{2\pi}{p}\left(D_{c}^{\scriptsize\mbox{UV}}Q_{c}+D_{\chi}^{\scriptsize\mbox{UV}}Q_{\chi}+D_{u}^{\scriptsize\mbox{UV}}Q_{u}\right)}{\cal
Z}_{\scriptsize\mbox{UV}}\,,$ (39)

while the IR partition function transforms as666$Q_{c}$ does not contribute to
the IR phase since the composites are color singlets.

$\displaystyle{\cal Z}_{\scriptsize\mbox{IR}}\longrightarrow
e^{i\frac{2\pi}{p}\left(D_{\chi}^{\scriptsize\mbox{IR}}Q_{\chi}+D_{u}^{\scriptsize\mbox{IR}}Q_{u}\right)}{\cal
Z}_{\scriptsize\mbox{IR}}\,,$ (40)

where
$D_{\chi}^{\scriptsize\mbox{IR}},D_{u}^{\scriptsize\mbox{IR}}\in\mathbb{Z}$
are group-theoretical coefficients that are chosen to match
$\mathbb{Z}_{p}^{d\chi}[U(1)_{A}]^{2}$ and
$\mathbb{Z}_{p}^{d\chi}[SU(n_{\chi})]^{2}$ anomalies. Since the UV-IR anomaly
matching is mod $p$, we must have777$\mathbb{Z}_{p}^{d\chi}$ is a good
symmetry in the color background, and thus we must have
$D_{c}^{\scriptsize\mbox{UV}}=p\ell_{c}$.

$\displaystyle D_{c}^{\scriptsize\mbox{UV}}=p\ell_{c}\,,\quad
D_{\chi}^{\scriptsize\mbox{UV}}-D_{\chi}^{\scriptsize\mbox{IR}}=p\ell_{\chi}\,,\quad
D_{u}^{\scriptsize\mbox{UV}}-D_{u}^{\scriptsize\mbox{IR}}=p\ell_{u}\,,$ (41)

for some $\ell_{c,\chi,u}\in\mathbb{Z}$. Thus, the ratio between the UV and IR
partition functions reads

$\displaystyle\frac{{\cal Z}_{\scriptsize\mbox{UV}}}{{\cal
Z}_{\scriptsize\mbox{IR}}}=e^{i2\pi\left(\ell_{c}Q_{c}+\ell_{\chi}Q_{\chi}+\ell_{u}Q_{u}\right)}\,,$
(42)

and the matching of the $\mathbb{Z}_{p}^{d\chi}[\mbox{CFU}]$ anomaly requires

$\displaystyle\ell_{c}Q_{c}+\ell_{\psi}Q_{\chi}+\ell_{u}Q_{u}\in\mathbb{Z}\,,$
(43)

for all allowed topological charges. Suppose no integers $\ell_{c,\chi,u}$
exist that satisfy this condition for a given allowed fractional topological
charges. In that case, the composites cannot solely match the
$\mathbb{Z}_{p}^{d\chi}[\mbox{CFU}]$ anomaly.

A minimal way out would be breaking
$\mathbb{Z}_{p}^{d\chi}\longrightarrow\mathbb{Z}_{q<p}$ via condensate
formation provided that the anomaly $\mathbb{Z}_{q}[\mbox{CFU}]$ vanishes.
Usually, a condensate would ordinarily break $SU(n_{\psi})$, $SU(n_{\chi})$,
and $U(1)_{A}$. Thus, one must postulate that all gauge-invariant condensates
charged under these symmetries have zero vacuum expectation values. Otherwise,
the condensation of such operators would oversaturate these anomalies, which
are assumed to be matched by composites. In addition, one needs to build a
neutral operator under the continuous symmetries, charged under
$\mathbb{Z}_{p}^{d\chi}$, and has a non-zero vacuum expectation value. If it
exists, such an operator would have a scaling dimension larger than the
vanishing lower-order condensates. Although this scenario cannot be ruled out,
we find it contrived in the examples of the $2$-index chiral theories we
discuss here.

This leaves us with the possibility that if condition (43) is violated, the
$\mathbb{Z}_{p}^{d\chi}[\mbox{CFU}]$ anomaly can be matched by a symmetry-
preserving topological quantum field theory (TQFT). In Cordova:2019bsd ;
Cordova:2019jqi , it was shown that the matching of
$\mathbb{Z}_{p}^{d\chi}$-gravitational anomalies by a unitary and symmetry-
preserving TQFT is obstructed on a spin manifold. This obstruction can also be
shown to hold in the case of $\mathbb{Z}_{p}^{d\chi}$[CFU] anomaly
Anber:2021iip .

We conclude that if condition (43) is violated, the theory probably cannot
flow to a phase with massless composites.

### (III) Spontaneous symmetry-breaking

In this scenario, the theory becomes strongly coupled; it confines (for $N$
even) and breaks its global symmetries spontaneously. We say that the theory
flows to a spontaneous symmetry-breaking (SSB) phase. An important aspect of
this work involves identifying the minimal set of condensates that break
global symmetries while matching the anomalies. These condensates break
$G^{\scriptsize\mbox{g}}$ down to $H\subset G^{\scriptsize\mbox{g}}$, with the
requirement that $H$ remains anomaly-free. Without satisfying this condition,
the symmetry breaking alone would not sufficiently match the UV anomaly. It is
possible for composite fermions to match a non-vanishing anomaly in $H$, but
it is crucial that these fermions do not undermine the matching of the
$G^{\scriptsize\mbox{g}}$ anomalies achieved by the condensates. Our focus did
not involve searching for composites that could match the anomalous unbroken
subgroups.

Generally, $H$ can be expressed as
$H=H^{c}\times\mathbb{Z}_{q_{1}}\times\mathbb{Z}_{q_{2}}$, where $H^{c}$
represents the continuous part of $H$, $\mathbb{Z}_{q_{1}}$ “collectively”
denotes the unbroken discrete subgroups of $SU(n_{\psi})\times
SU(n_{\chi})\times U(1)_{A}$, and $\mathbb{Z}_{q_{2}}$ represents the unbroken
subgroup of $\mathbb{Z}_{p}^{d\chi}$. If the condensates leave a discrete
subgroup unbroken, we must examine its anomalies. In addition, if the theory
possesses a $1$-form/$0$-form mixed anomaly, there can be fractionalization
classes, and hence, an ambiguity in calculating the cubic discrete anomalies
Delmastro:2022pfo . One must ensure the condensates do not leave any discrete
anomaly in any fractionalization class888We would like to thank Erich Poppitz
for illuminating discussions about this point.. In a few examples, we observe
that lower-order condensates (such as the $2$-fermion condensates) lead to
anomalous unbroken discrete subgroups. Consequently, the formation of other
(higher-order) condensates becomes necessary to break the symmetries into non-
anomalous subgroups.

In strongly-coupled theories, it is generally believed that higher-order
bosonic operators undergo condensation. In this work, through anomaly matching
conditions, we provide kinematical reasons behind this condensation.

Now we turn our attention to the matching of CFU anomalies. Given the better
understanding of the nature of the unbroken discrete subgroups of $U(1)_{A}$
and their anomalies, here, we provide a more in-depth discussion of the CFU
anomalies than the earlier work Anber:2021iip . As mentioned above, we
encounter two types of such anomalies: $U(1)_{A}[\mbox{CFU}]$ and
$\mathbb{Z}_{p}^{d\chi}[\mbox{CFU}]$ anomalies. In all the examples we have
examined, we consistently observe the trivialization of
$\mathbb{Z}_{p}^{d\chi}[\mbox{CFU}]$ by the condensates, which break the
$U(1)_{A}$ symmetry. On the other hand, the matching of the
$U(1)_{A}[\mbox{CFU}]$ anomaly through condensates is a more intricate process
that required closer examination.

As emphasized earlier, the color topological charge does not play a role in
this particular anomaly. Consequently, we can view it as an anomaly of
$U(1)_{A}$ in the presence of the flavor center and $U(1)_{A}$ fluxes. In the
examples we have studied, the condensates break the flavor center, rendering
this anomaly irrelevant. In simpler terms, the full breaking of the flavor
center automatically matches the $U(1)_{A}[\mbox{CFU}]$ anomaly. This can be
understood through the following principle: if a triangle (anomaly) diagram
involves three abelian symmetries, namely $G_{1}$, $G_{2}$, and $G_{3}$ (in
this case, $G_{1}$ through $G_{3}$ are the abelian discrete groups
corresponding to turning on the CFU fluxes), the complete breaking of at least
one of these symmetry groups will resolve the anomaly.

To extract more valuable insights from this anomaly, we can focus our
attention solely on the color-$U(1)_{A}$ fluxes by deactivating the flavor
background. In doing so, the $U(1)_{A}[\mbox{CU}]$ anomaly becomes a mixed
anomaly of $U(1)_{A}$ in the presence of fractional $U(1)_{A}$ flux (keeping
in mind that the $U(1)_{A}$ flux still needs to combine with the color flux to
satisfy the cocycle conditions. Yet, the color topological charge remains
uninvolved in the anomaly). Superficially, one might consider this to be
equivalent to the $[U(1)_{A}]^{3}$ anomaly. However, this is not the case
since the latter anomaly only encompasses integer fluxes of $U(1)_{A}$,
whereas the $U(1)_{A}[\mbox{CU}]$ anomaly incorporates the minimal flux of
$U(1)_{A}$. Consequently, the latter is more restrictive in nature compared to
the $[U(1)_{A}]^{3}$ anomaly. Let the discrete flux of $U(1)_{A}$ be a
$\mathbb{Z}_{n}$ flux, and let a particular condensate break $U(1)_{A}$ to
$\mathbb{Z}_{m}\supseteq\mathbb{Z}_{n}$. Then the $U(1)[\mbox{CU}]$ anomaly
can be thought of as a $\mathbb{Z}_{m}[\mathbb{Z}_{n}]^{2}$ anomaly, which can
be checked via (32). If $\mathbb{Z}_{m}\subset\mathbb{Z}_{n}$ and the anomaly
$[\mathbb{Z}_{m}]^{3}$ vanishes, the breaking of $U(1)_{A}$ to
$\mathbb{Z}_{m}$ automatically matches the $U(1)[\mbox{CU}]$ anomaly as the
symmetry corresponding to the discrete flux of $U(1)_{A}$ is broken.

Next, we discuss the condensates that cause the symmetries to break. A gauge-
invariant condensate is a bosonic operator

$\displaystyle{\cal C}=\psi^{\alpha_{\psi}}\chi^{\alpha_{\chi}}\,,$ (44)

and $\alpha_{\psi}$ and $\alpha_{\chi}$ satisfy the conditions

$\displaystyle 2\alpha_{\psi}+(N-2)\alpha_{\chi}\in
N\mathbb{Z}^{+}\,,\quad\alpha_{\psi}+\alpha_{\chi}\in 2\mathbb{Z}^{+}\,.$ (45)

One needs as many condensates as necessary to break $G^{\scriptsize\mbox{g}}$
to an anomaly-free subgroup. Distinct condensates will break
$G^{\scriptsize\mbox{g}}$ down to $H_{1}\subset G^{\scriptsize\mbox{g}}$,
$H_{2}\subset G^{\scriptsize\mbox{g}}$, etc. If the subgroups
$\\{H_{1},H_{2},..\\}$ do not share common generators,
$G^{\scriptsize\mbox{g}}$ will break to unity. Finding the breaking patterns
of a group $G^{\scriptsize\mbox{g}}$ because of the condensation of single or
many operators transforming in the defining or higher-dimensional
representations of $G^{\scriptsize\mbox{g}}$ is, in general, a complicated
problem. Only a few cases have been discussed in the literature; see, e.g.,
Li:1973mq ; Elias:1975yd ; Wu:1981eb ; Ruegg:1980gf ; Jetzer:1983ij  and
references therein. The question then is, how many condensates does the theory
develop in the IR? There is no known answer to this question. However, there
must be at least as many condensates as needed to match all anomalies.

###  3.2 Minimizing the IR degrees of freedom

Beyond ’t Hooft anomalies, are there additional sources of information that
can be harnessed to make conjectures about the infrared (IR) phase of a
strongly coupled theory? In Appelquist:1999hr ; Appelquist:1999vs ;
Appelquist:2000qg , a constraint on the structure of strongly coupled
asymptotically-free field theories was proposed. The constraint is an
inequality favoring an IR phase with fewer degrees of freedom (DOF). It was
also proposed to use the free energy to characterize DOF. The effective
degrees of freedom ${\cal A}$ of free $n_{B}$ massless real scalars and free
$n_{f}$ massless Weyl fermions are given in terms of the free energy density
$F$ as ($T$ is an infinitesmal temperature)

$\displaystyle{\cal
A}\equiv\frac{90F}{\pi^{2}T^{4}}=n_{B}+\frac{7}{4}n_{f}\,.$ (46)

First, we may use (46) to favor between a phase of composite fermions or a
phase with spontaneous symmetry breaking (SSB). As we pointed out above, we
could not find composite fermions that matched the anomalies. Yet, one may be
tempted to use (46) to predict whether the theory flows to an IR CFT. In a
weakly-coupled CFT, the IR DOF are the same UV DOF. On the other hand, the DOF
in a spontaneously broken phase, assuming the global symmetry $SU(N_{f})$
entirely breaks, are $N_{f}^{2}-1$ Goldstones999Notice that a theory that
fully breaks its global symmetries will match its ’t Hooft anomalies in the
IR. We assume that enough condensates form to obey the matching conditions..
Let us define $\Delta{\cal A}$ as the difference between the DOF in the two
scenarios. Then, we have

$\displaystyle\Delta{\cal
A}=\underbrace{n_{\psi}^{2}+n_{\chi}^{2}-2}_{\mbox{Goldstones}}-\left\\{\underbrace{2(N^{2}-1)}_{\mbox{gluons
DOF}}+\frac{7}{4}\left(n_{\psi}\mbox{dim}_{\psi}+n_{\chi}\mbox{dim}_{\chi}\right)\right\\}\,.$
(47)

According to the conjecture, a theory with $\Delta{\cal A}>0$ disfavors an SSB
phase. It can be easily checked that all the theories in Table 1 yield
$\Delta{\cal A}<0$, favoring a phase with broken symmetries. This is to be
expected since $n_{\psi},n_{\chi}\sim N$ and
$\mbox{dim}_{\psi},\mbox{dim}_{\chi}\sim N^{2}$. Thus, while the SSB phase has
$\sim N^{2}$ DOF, a phase with CFT has $\sim N^{3}$ DOF. Then, one may naively
conclude that all theories in Table 1 will break their symmetries and flow to
a Goldstone phase. This conclusion, however, totally ignores the dynamics of
the theory on the way from UV to IR. A theory must enter a strongly-coupled
regime to form condensates and break its continuous symmetries, i.e., breaking
the symmetries has to happen dynamically since no elementary scalars exist. As
we argued above, some of our theories have robust IR fixed points at weak
coupling, indicating that it is most unlikely they can form condensates.
Consequently, in the subsequent analysis, we avoid employing the
aforementioned hypothesis to favor between an SSB or CFT phase. Instead, we
use the $\beta$-function analysis to check whether a theory flows to an IR
CFT101010This method was used in Anber:2023urd  to predict the IR phase of a
theory with a single adjoint and $N_{f}$ fundamental flavors of Weyl fermions.
It was found that the $\Delta{\cal A}$ calculations are consistent with the
prediction of perturbative $\beta$-function. The fact that this analysis does
not hold for the $2$-index chiral theories is attributed to the large number
of degrees of freedom of a CFT, which always exceeds the number of degrees of
freedom of an SSB phase..

However, assuming the existence of multiple sets of condensates, each capable
of accounting for all observed anomalies via SSB, we can employ the
aforementioned line of reasoning to make a prediction. Presumably, the set of
condensates that causes the flavor group to break into the largest subgroup
will be preferred due to its associated reduction in the number of infrared
degrees of freedom.

The following sections are devoted to systematically applying the above ideas
to the concrete theories in Table 1. We start our discussion by working out
all the details. As we progress through the list of theories, we build on the
previous experience and shorten our discussion.

##  4 Fermionic theories

This section systematically studies theories that admit fermionic operators in
their spectrum. These are $(N=5,k=1)$, $(N=6,k=2)$, $(N=6,k=1)$, and
$(N=10,k=2)$. Our analysis indicates that the first two theories form
condensates and break their global symmetries, while the last two flow to a
CFT.

###  4.1 $SU(5),k=1$

This theory admits a single Weyl fermion $\psi$ and $n_{\chi}=9$ flavors of
$\chi$ Weyl fermions. In addition, we have
$r=\mbox{gcd}(n_{\psi}T_{\psi},n_{\chi}T_{\chi})=\mbox{gcd}(7,27)=1$,
indicating that the theory does not possess a discrete chiral symmetry. The
solutions to the cocycle conditions (14) give
$\mathbb{Z}_{5}\times\mathbb{Z}_{9}$ as the discrete division group. Thus, the
faithful global symmetry is

$\displaystyle G^{\scriptsize\mbox{g}}=\frac{SU(9)_{\chi}\times
U(1)_{A}}{\mathbb{Z}_{5}\times\mathbb{Z}_{9}}\,,$ (48)

and the $U(1)_{A}$ charges of $\psi$ and $\chi$ are

$\displaystyle q_{\psi}=-27\,,\quad q_{\chi}=7\,.$ (49)

Since both $q_{\psi}$ and $q_{\chi}$ are odd,
$(-1)^{F}\equiv\mathbb{Z}_{2}^{F}$ fermion-number symmetry, which acts on
$(\psi,\chi)$ as $(\psi,\chi)\longrightarrow-(\psi,\chi)$, is a subgroup of
$U(1)_{A}$.

The topological charges of the CFU fluxes are given by:

$\displaystyle Q_{c}=\frac{4m^{2}}{5}\,,m\in\mathbb{Z}_{5}\,,\quad
Q_{\chi}=\frac{8p^{\prime 2}}{9}\,,p^{\prime}\in\mathbb{Z}_{9}\,,\quad
Q_{u}=s^{2}\,,s\in\mathbb{Z}_{45}\,,$ (50)

and $(m,p^{\prime},s)$ are chosen to satisfy (14). The theory admits a set of
anomalies listed in Table 2 (from here on, we give the phase of the
corresponding anomaly).

Anomaly | Equation | Value
---|---|---
$[U(1)_{A}]^{3}$ | $\kappa_{u}^{3}=n_{\psi}q_{\psi}^{3}\dim\psi+n_{\chi}q_{\chi}^{3}\dim\chi$ | $-264375$
$U(1)_{A}[SU(9)_{\chi}]^{2}$ | $q_{\chi}\dim\chi$ | $70$
$\left[SU(9)_{\chi}\right]^{3}$ | $\dim\chi$ | $10$
$U(1)_{A}[\text{grav}]$ | $2(n_{\psi}q_{\psi}\dim\psi+n_{\chi}q_{\chi}\dim\chi)$ | $450$
$U(1)_{A}[\text{CFU}]$ | $q_{\chi}\dim\chi\,Q_{\chi}+\kappa_{u^{3}}Q_{u}$ | $\frac{560}{9}p^{\prime 2}-264375s^{2}$
Table 2: Anomalies of $SU(5),k=1$.

Notice that, as pointed out above, the $U(1)_{A}[\text{CFU}]$ anomaly does not
depend on the color topological charge. We can also put the theory on
$\mathbb{CP}^{2}$ by employing fluxes in the centers of $SU(5)$ and
$SU(9)_{\chi}$ accompanied by a $U(1)_{A}$ flux, as can be easily checked from
(19).

The $2$-loop and $3$-loop $\beta$-function analysis show that the theory has
an IR fixed point at somewhat large coupling-constant:
$\frac{g_{*}^{2}}{4\pi}\approx 0.64$ and $\frac{g_{*}^{2}}{4\pi}\approx 0.34$,
respectively. Therefore, such a fixed point is not robust. We conclude that
either the theory forms composite fermions or flows to an SSB phase.

### Matching by composites

We used the systematic approach discussed in Section 3 to search for a set of
composite fermions. We found a pair of operators

$\displaystyle{\cal F}_{1}=\psi\chi^{6}\,,\quad{\cal
F}_{2}=\psi^{7}\chi^{22}\,,$ (51)

with ${\cal N}_{1}=36$ and ${\cal N}_{2}=9$ copies that matched the
$\left[U(1)_{A}\right]^{3}$ and $U(1)_{A}[\mbox{grav}]$ anomalies. Yet, this
pair failed to match the $U(1)_{A}\left[SU(9)_{\psi}\right]^{2}$ anomaly. The
upper bound on the number of the IR fermion species is ${\cal N}\sim 132$. The
large number of partitions of ${\cal N}$ is ${\cal O}(10^{7})$, which hindered
the abilities of our search algorithm. We failed to find a set of composites
that matches the full set of anomalies.

### Matching by condensates

We now turn to the formation of condensates. The lowest-order condensate is

$\displaystyle{\cal
C}_{1}^{i}=\epsilon^{a_{1}a_{2}a_{3}a_{4}a_{5}}\epsilon_{\alpha_{1}\alpha_{2}}\psi_{(a_{1}a_{2})}^{\alpha_{1}}\chi_{[a_{3}a_{4}a_{5}]}^{\alpha_{2},\,i}\,,$
(52)

where $a_{1},..,a_{5}$ are color, $\alpha_{1},\alpha_{2}$ are spinor, and $i$
is a $SU(9)_{\chi}$ flavor indices. This condensate vanishes identically owing
to the symmetrizing over $a_{1},a_{2}$. Yet, one can evade this problem by
inserting gauge-covariant gluonic fields
$(f_{\mu\nu}^{c})_{a_{i}}^{a_{j}}\sigma^{\mu\nu}$:

$\displaystyle{\cal C}_{1}^{i}\longrightarrow\tilde{\cal
C}_{1}^{i}=\epsilon^{a_{1}a_{2}a_{3}a_{4}a_{5}}\epsilon_{\alpha_{1}\alpha_{2}}(f_{\mu\nu}^{c})_{a_{2}}^{a_{6}}\sigma^{\mu\nu}\psi_{(a_{1}a_{6})}^{\alpha_{1}}\chi_{[a_{3}a_{4}a_{5}]}^{\alpha_{2},\,i}\,.$
(53)

This trick will always be followed whenever the statistics of indices cause
some operator to vanish. ${\cal C}_{1}^{i}$ transforms in the defining
representation of $SU(9)_{\chi}$, and thus, it breaks it down to $SU(8)$.
However, the condensation of ${\cal C}_{1}^{i}$ leaves a $U(1)$ generator of
$SU(9)\times U(1)_{A}$ unbroken. To see that, we go to a basis where ${\cal
C}_{1}^{i}\propto\delta_{i,9}$. In this basis, the unbroken $SU(8)$ group acts
on the $8\times 8$ upper block matrices of the original $9\times 9$ unitary
matrices of $SU(9)$. Now, it is easy to see that the $SU(9)$ Cartan generator
$H_{8}=\mbox{diag}\left(1,1,...,-8\right)$ combines with the $U(1)_{A}$
generator to leave the vacuum $\delta_{i,9}$ invariant:

$\displaystyle
e^{i2\pi(-20\beta)}\left[\begin{array}[]{ccc}e^{i2\pi\alpha}&0...&0\\\
...&...&...\\\
0&...&e^{i2\pi(-8\alpha)}\end{array}\right]\left[\begin{array}[]{c}0\\\ ...\\\
1\end{array}\right]=\left[\begin{array}[]{c}0\\\ ...\\\
1\end{array}\right]\,,$ (63)

where $\alpha$ and $\beta$ are the Cartan and $U(1)_{A}$ phases, respectively.
The direction $2\alpha=-5\beta$ is the unbroken $U(1)$ direction. The unbroken
$SU(8)$ has a non-vanishing cubic anomaly. In addition, the unbroken $U(1)$
symmetry inherits the $U(1)_{A}[\mbox{grav}]$ anomaly, signaling that such
breaking is incomplete or inconsistent with the anomaly-matching conditions.

Another condensate is (we suppress color and spinor indices to reduce clutter)

$\displaystyle{\cal C}_{2}^{(ij)}=\psi^{2}\chi^{(i}\chi^{j)}\,,$ (64)

which transforms in the $2$-index symmetric representation of the flavor
group111111We also insert gluons if the statistics cause the condensate to
vanish.. The general form of the “Higgs” potential of the condensate is

$\displaystyle V({\cal C}_{2})=-\frac{1}{2}\mu^{2}{\cal C}_{2}^{(ij)}{\cal
C}_{2(ij)}+\frac{1}{4}\lambda_{1}({\cal C}_{2}^{(ij)}{\cal
C}_{2(ij)})^{2}+\frac{1}{4}\lambda_{2}({\cal C}_{2(ij)}{\cal
C}_{2}^{(jk)}{\cal C}_{2(kl)}{\cal C}^{(li)}_{2})\,,$ (65)

for some real parameters $\mu^{2}>0$, $\lambda_{1}$, and $\lambda_{2}$. In the
case $\lambda_{2}>0$, the condensate has a non-zero vacuum expectation value
and we can pick the form of the condensate to be ${\cal C}_{2}\propto I_{9}$
Li:1973mq . This breaks $SU(9)$ to the anomaly-free subgroup $SO(9)$.

Is there a combination of $SU(9)_{\chi}\times U(1)_{A}$ that breaks to a
remaining $U(1)$ symmetry? As $U(1)_{A}$ is abelian, we need to consider the
subgroup generated by the Cartan subalgebra of $SU(9)$. The “unnormalized”
generators of the Cartan subalgebra of $SU(9)$ are:

$\displaystyle[H_{m}]_{ij}=\sum_{k=1}^{m}\delta_{ik}\delta_{jk}-m\delta_{i,m+1}\delta_{j,m+1}\,,\quad
m=1,2,...,8\,.$ (66)

A general $SU(9)$ element generated by the Cartan subalgebra has the form
$\exp(2\pi i\alpha_{m}H_{m})$, $m=1,2,..,8$ and $\alpha_{m}\in\mathbb{[}0,1)$.
A combined $SU(9)\times U(1)_{A}$ transformation acts on ${\cal C}^{ij}$ via:

$\displaystyle{\cal C}^{\prime(ij)}_{2}=e^{2\pi i(-40\beta)}\left(e^{2\pi
i\alpha_{m}H_{m}}\right)^{ik}\left(e^{2\pi i\alpha_{m}H_{m}}\right)^{jl}{\cal
C}_{2(kl)}\,,$ (67)

and should leave the vacuum expectation value invariant. Thus, we need

$\displaystyle e^{2\pi i(-40\beta)}\left(e^{2\pi
i\alpha_{m}H_{m}}\right)^{ik}\left(e^{2\pi
i\alpha_{m}H_{m}}\right)^{jl}I_{9}=I_{9}\,.$ (68)

It can be easily checked that there are no nontrivial solutions to the above
equation, indicating that no $U(1)$ direction is left unbroken.

Under the action of $U(1)_{A}$, the condensate transforms as ${\cal
C}^{(ij)}_{2}=\psi^{2}\chi^{(i}\chi^{j)}\longrightarrow{\cal
C}^{\prime(ij)}_{2}=e^{i2\pi(-40\beta)}\psi^{2}\chi^{(i}\chi^{j)}\,,$ where
$\beta\in[0,1)$ is the $U(1)_{A}$ parameter. So it appears that the condensate
is invariant under a discrete $\mathbb{Z}_{40}$ subgroup of $U(1)_{A}$. But
recall that the global symmetry group includes a division by the
$\mathbb{Z}_{5}$ center of the color group. $\mathbb{Z}_{5}$ is not a subgroup
of $SU(9)$, therefore it can only quotient $U(1)_{A}$, so the parameter
$\beta$ is in fact a $U(1)_{A}/\mathbb{Z}_{5}$ parameter and
$\beta\in[0,1/5)$. Therefore the condensate only exhibits an unbroken
$\mathbb{Z}_{8}$ symmetry.

The discrete symmetry $\mathbb{Z}_{8}$ has non-perturbative anomalies, as can
easily be checked using (29) and (30), meaning that the condensation of ${\cal
C}_{2}^{(ij)}$ is insufficient to match the full set of anomalies. Notice that
since both $\psi$ and $\chi$ have odd charges under $U(1)_{A}$, any unbroken
discrete subgroup of $U(1)_{A}$ necessarily contains $(-1)^{F}$, and thus, we
can use the twisted group $\mbox{Spin}^{\mathbb{Z}_{2m}}$ to detect the
nonperturbative anomaly as given from (30). Moreover, since the theory does
not possess a $1$-form symmetry, there is no ambiguity in calculating the
discrete-symmetry anomaly Delmastro:2022pfo .

In searching for a condensate that does not leave behind a non-anomalous
$U(1)$ or discrete subgroup, we consider the most general bosonic operator:

$\displaystyle{\cal C}=\psi^{\alpha_{\psi}}\chi^{\alpha_{\chi}}\,,\quad
2\alpha_{\psi}+3\alpha_{\chi}\in
5\mathbb{Z}^{+}\,,\alpha_{\psi}+\alpha_{\chi}\in 2\mathbb{Z}^{+}\,.$ (69)

This condensate carries a charge of $-27\alpha_{\psi}+7\alpha_{\chi}$ under
$U(1)_{A}$, and thus, breaks $U(1)_{A}$ down to
$\mathbb{Z}_{(-27\alpha_{\psi}+7\alpha_{\chi})/5}$. We used both (29) and (30)
to check the nonperturbative anomalies of
$\mathbb{Z}_{(-27\alpha_{\psi}+7\alpha_{\chi})/5}$ and found that both
untwisted and twisted backgrounds yield the same results. The lowest-
dimensional condensate that breaks $U(1)_{A}$ to a non-anomalous subgroup has
$\alpha_{\psi}=1$ and $\alpha_{\chi}=11$. In this case, $\mathbb{Z}_{10}$ is
the anomaly-free subgroup.

The condensate

$\displaystyle{\cal C}_{3}=\psi\chi^{11}$ (70)

transforms in a higher representation of $SU(9)_{\chi}$. One can contract $9$
out of the $11$ flavor indices of ${\cal C}_{3}$ with the Levi-Civita tensor,
leaving $2$ free indices. Then, we can rearrange the free indices (possibly
with insertions of gluons in case the statistics cause the condensate to
vanish) such that ${\cal C}_{3}$ transforms in the $2$-index symmetric
representation of $SU(9)$:

$\displaystyle{\cal C}_{3}^{(ij)}=\psi\chi^{9}\chi^{(i}\chi^{j)}\,.$ (71)

The condensing of ${\cal C}_{3}^{(ij)}$ breaks $\frac{SU(9)_{\chi}\times
U(1)_{A}}{\mathbb{Z}_{5}\times\mathbb{Z}_{9}}$ down to the anomaly-free
subgroup $SO(9)\times\mathbb{Z}_{10}$.

Alternatively, one can search for a companion condensate to ${\cal
C}_{2}^{(ij)}$ that breaks $U(1)_{A}$ to a discrete subgroup $\mathbb{Z}_{q}$,
such that $\mbox{gcd}(q,8)=2$. This ensures that the formation of these two
condensates breaks $U(1)_{A}$ down to the anomaly-free subgroup
$\mathbb{Z}_{2}^{F}$, which is the fermion number. The companion condensate
with the lowest dimension is ${\cal C}_{4}=\chi^{10}$, which, superficially,
breaks $U(1)_{A}$ down to $\mathbb{Z}_{q}=\mathbb{Z}_{14}$. This, however, is
an immature conclusion. One can contract $9$ flavor indices of ${\cal C}_{4}$
with a Levi-Civita tensor leaving one free index. Then, ${\cal C}_{4}$
transforms in the fundamental representation of $SU(9)$, and according to the
discussion preceding (63), it breaks it down to $SU(8)\times U(1)$. Because of
the unbroken $U(1)$ generator, the condensation of ${\cal C}_{2}$ along with
${\cal C}_{4}$ break $\frac{SU(9)_{\chi}\times
U(1)_{A}}{\mathbb{Z}_{5}\times\mathbb{Z}_{9}}$ down to a subgroup that
contains the anomalous $\mathbb{Z}_{8}$. More than this is needed to match the
full set of anomalies.

We might continue searching for suitable condensates that break
$G^{\scriptsize\mbox{g}}$ to an anomaly-free subgroup. However, the lesson
from the above discussion is that it is generically a complex exercise.

Since $SO(9)$ is the largest anomaly-free subgroup of $SU(9)$, the
condensation of ${\cal C}_{3}^{(ij)}$ leads to the smallest number of the IR
Goldstones, and hence, we predict that the theory will flow to a phase with
the global symmetry broken down to the anomaly-free
$SO(9)\times\mathbb{Z}_{10}$. This is the minimal scenario. However, because
of strong dynamics, nothing forbids the theory from forming all kinds of
condensates, breaking $G^{\scriptsize\mbox{g}}$ down to the anomaly-free
$\mathbb{Z}_{2}^{F}$ fermion number symmetry.

In an equally alternative scenario, $SU(9)$ could be broken down to the
anomaly-free $Sp(8)$ by a condensate transforming in the $2$-index anti-
symmetric representation. However, since the dimensions of $Sp(8)$ and $SO(9)$
are identical, anomalies and the argument of the number of Godstones cannot
distinguish between the two possible symmetry-breaking scenarios. In general,
a condensate transforming in the $2$-index symmetric representation of
$SU(2N+1)$ breaks this group down to $SO(2N+1)$, while a condensate in the
$2$-index anti-symmetric representation breaks it down to $Sp(2N)$. Both
$SO(2N+1)$ and $Sp(2N)$ have dimension $N(2N+1)$.

Interestingly, the operator ${\cal C}_{3}^{(ij)}$ has a scaling dimension of
at least $15$ (it could have a higher dimension if gluon fields are needed to
avoid the vanishing of the condensate because of fermi-statistics). That such
condensate with a large-scaling dimension must condense in the IR to match the
complete set of anomalies is remarkable. Generally, it is natural to expect
that a strongly coupled theory forms higher-order condensates. In this
example, however, this formation is not a question about the dynamics; rather,
it is a necessary condition for the theory to obey the kinematical constraints
imposed by anomalies.

Does our proposed condensate ${\cal C}_{3}^{(ij)}$ match the
$U(1)_{A}[\mbox{CFU}]$ anomaly? The answer is affirmative. ${\cal
C}_{3}^{(ij)}$ breaks $SU(9)_{\chi}$ flavor group down to $SO(9)$. The latter
does not have a center symmetry, while the former group has a $\mathbb{Z}_{9}$
center. Thus, we conclude that the condensate breaks $\mathbb{Z}_{9}$
maximally, matching the $U(1)_{A}[\mbox{CFU}]$ anomaly. Next, we may turn off
the flavor background, restricting ourselves to the color center and
$U(1)_{A}$ (CU) fluxes. In this case, we have
$(m,p^{\prime},s)=(1,0,\frac{1}{5})$, and keeping in mind that the CU anomaly
does not depend on the color topological charge, we find that this is an
anomaly of the axial current in the background of a $\mathbb{Z}_{5}$ flux. The
condensation of ${\cal C}_{3}$ breaks $U(1)_{A}$ to $\mathbb{Z}_{10}$. Thus,
the $U(1)_{A}[\mbox{CU}]$ anomaly becomes the
$\mathbb{Z}_{10}[\mathbb{Z}_{5}]^{2}$ anomaly discussed around Eq. (32).
However, from the last line in Table 2, the anomaly coefficient becomes,
$\frac{-264375}{5^{2}}=-10575$, which is $0$ modulo $5$. Therefore, in this
case, the anomaly becomes trivial, and the $U(1)_{A}[\mbox{CU}]$ anomaly is
automatically matched.

###  4.2 $SU(6),k=2$

This theory has a single $\psi$ Weyl fermion along with $5$ flavors of $\chi$
fermions. Thus, the continuous global symmetry is $SU(5)_{\chi}\times
U(1)_{A}$. The charges of $\psi$ and $\chi$ under $U(1)_{A}$ are

$\displaystyle q_{\psi}=-5\,,\quad q_{\chi}=2\,.$ (72)

Owing to the fact
$r=\mbox{gcd}(n_{\psi}T_{\psi},n_{\chi}T_{\chi})=\mbox{gcd}(8,20)=4$, the
theory is also endowed with a $\mathbb{Z}_{4}^{d\chi}$ chiral symmetry, which
is taken to act on $\chi$ with a unit charge. It can be checked that this is a
genuine symmetry since neither $\mathbb{Z}_{4}$ nor a subgroup of it can be
absorbed in rotations in the centers of $SU(6)\times SU(5)_{\chi}$. To show
that, we try to absorb the elements $e^{i\frac{2\pi\ell}{4}}$, $\ell=1,2,3$,
in the centers of $SU(6)\times SU(5)_{\chi}$:

$\displaystyle\mathbb{Z}_{4}:\quad\psi\longrightarrow e^{2\pi
i\frac{2m}{6}}\psi=\psi\,,\quad\chi\longrightarrow e^{-2\pi
i\frac{2m}{6}}e^{-2\pi i\frac{p^{\prime}}{5}}\chi=e^{2\pi
i\frac{l}{4}}\chi\,.$ (73)

No values of $m$ and $p^{\prime}$ satisfy these equations for $\ell=1,2,3$,
and therefore, $\mathbb{Z}_{4}^{d\chi}$ is a genuine symmetry. An identical
procedure is employed in the rest of the theories to ascertain the genuineness
of discrete chiral symmetries.

To determine the faithful global symmetry, we must find the quotient group by
solving the consistency conditions (14). This gives
$\mathbb{Z}_{3}\times\mathbb{Z}_{5}$ as the group we divide by. Putting
everything together and remembering that the theory possesses a
$\mathbb{Z}_{2}^{[1]}$ $1$-form center symmetry, we write the faithful global
group:

$\displaystyle G^{\scriptsize\mbox{g}}=\frac{SU(5)_{\chi}\times
U(1)_{A}}{\mathbb{Z}_{3}\times\mathbb{Z}_{5}}\times\mathbb{Z}_{4}^{d\chi}\times\mathbb{Z}_{2}^{(1)}\,.$
(74)

The $\mathbb{Z}_{2}^{F}$ fermion number symmetry is contained in the
generators of the product group $U(1)_{A}\times\mathbb{Z}_{4}^{d\chi}$ (notice
that the $U(1)_{A}$ charges of $\psi$ and $\chi$ are odd and even,
respectively)

$\displaystyle\mathbb{Z}_{2}^{F}\subset
U(1)_{A}:\psi\longrightarrow-\psi\,,\quad\chi\longrightarrow\chi\,,$
$\displaystyle\mathbb{Z}_{2}^{F}\subset\mathbb{Z}_{4}^{d\chi}:\psi\longrightarrow\psi\,,\quad\chi\longrightarrow-\chi\,.$
(75)

The topological charges of the CFU fluxes are given by:

$\displaystyle Q_{c}=\frac{5m^{2}}{6}\,,m\in\mathbb{Z}_{3}\,,\quad
Q_{\chi}=\frac{4p^{\prime 2}}{5}\,,p^{\prime}\in\mathbb{Z}_{5}\,,\quad
Q_{u}=s^{2}\,,s\in\mathbb{Z}_{15}\,,$ (76)

and $(m,p^{\prime},s)$ are chosen to satisfy (14). The anomalies of the theory
are listed in Table 3.

Anomaly | Equation | Value
---|---|---
$[U(1)_{A}]^{3}$ | $\kappa_{u^{3}}=n_{\psi}q_{\psi}^{3}\dim\psi+n_{\chi}q_{\chi}^{3}\dim\chi$ | $-2025$
$U(1)_{A}[SU(5)_{\chi}]^{2}$ | $q_{\chi}\dim\chi$ | $30$
$\left[SU(5)_{\chi}\right]^{3}$ | $\dim\chi$ | $15$
$\mathbb{Z}_{4}^{d\chi}\left[U(1)_{A}\right]^{2}$ | $\kappa_{zu^{2}}=n_{\chi}q_{\chi}^{2}\dim\chi$ | $300\,\mbox{mod}\,4$
$\mathbb{Z}_{4}^{d\chi}\left[SU(5)_{\chi}\right]^{2}$ | $\dim\chi$ | $15\,\mbox{mod}\,4$
$U(1)_{A}[\text{grav}]$ | $2(n_{\psi}q_{\psi}\dim\psi+n_{\chi}q_{\chi}\dim\chi)$ | $90$
$\mathbb{Z}_{4}^{d\chi}[\text{grav}]$ | $2n_{\chi}\dim\chi$ | $150\,\mbox{mod}\,8$
$[\mathbb{Z}_{4}^{d\chi}]^{3}$ | (29) | $2250\,\mbox{mod}\,24$
$U(1)_{A}[\text{CFU}]$ | $q_{\chi}\dim\chi\,Q_{\chi}+\kappa_{u^{3}}Q_{u}$ | $24p^{\prime 2}-2025s^{2}$
$\mathbb{Z}_{4}^{d\chi}[\text{CFU}]$ | $n_{\chi}T_{\chi}Q_{c}+\dim\chi\,Q_{\chi}+\kappa_{zu^{2}}Q_{u^{2}}$ | $\frac{50}{3}m^{2}+12p^{\prime 2}+300s^{2}$
Table 3: Anomalies of $SU(6),k=2$.

It is worth noting that both $\mathbb{Z}_{4}^{d\chi}[\text{grav}]$ and
$\mathbb{Z}_{4}^{d\chi}[\text{CFU}]$ anomalies give at most a $\mathbb{Z}_{2}$
phase. Also, this theory cannot be put on $\mathbb{CP}^{2}$, as there are no
solutions to the conditions (19).

We first comment on the possibility that the theory flows to a Banks-Zaks
fixed point in the IR. The $2$-loop beta function of this theory gives
$\frac{g_{*}^{2}}{4\pi}\approx 8.5\gg 1$. This value of the coupling constant
is too large for perturbation theory to hold. At $3$-loops, we obtain
$\frac{g_{*}^{2}}{4\pi}\approx 0.73\,.$ Also, this coupling-constant value is
large, so we cannot conclude that our theory flows to a conformal fixed point
in the IR. In the following, we examine the possibilities of fermion
composites and SSB.

### Matching by composites

Here, we follow the argument in Section 3 to show that composite fermions
cannot solely match all the UV anomalies. The UV
$\mathbb{Z}_{4}^{d\chi}[\text{CFU}]$ anomaly of this theory is (unlike the
$U(1)_{A}[\text{CFU}]$ anomaly, it is important to notice that the color flux
contributes to the $\mathbb{Z}_{4}^{d\chi}[\text{CFU}]$ anomaly)

$\displaystyle
n_{\chi}T_{\chi}Q_{c}+\dim\chi\,Q_{\chi}+\kappa_{zu^{2}}Q_{u^{2}}=\frac{50}{3}m^{2}+12p^{\prime
2}+300s^{2}\,.$ (77)

In the IR, a set of gauge invariant composite fermions would generate the
corresponding $\mathbb{Z}_{4}^{d\chi}[\text{CFU}]$ anomaly:

$\displaystyle D_{\chi}^{\text{IR}}Q_{\chi}+D_{u}^{\text{IR}}Q_{u}$ (78)

for integers $D_{\chi}^{\text{IR}}$ and $D_{u}^{\text{IR}}$. The integers
$D_{\chi}^{\text{IR}}$ and $D_{u}^{\text{IR}}$ are group-theoretical
coefficients that are assumed to be found by matching all anomalies of
continuous symmetries. In the presence of a CFU background flux, the ratio
between the UV and IR partition functions after undergoing a
$\mathbb{Z}_{4}^{d\chi}$ transformation is given by:

$\displaystyle\frac{\mathcal{Z}^{\text{UV}}}{\mathcal{Z}^{\text{IR}}}=e^{\frac{i2\pi}{4}\left(\frac{50}{3}m^{2}+\left(12-D_{\chi}^{\text{IR}}\right)p^{2}+\left(300-D_{u}^{\text{IR}}\right)s^{2}\right)}=e^{\frac{i2\pi}{4}\left(\frac{50}{3}m^{2}+d_{\chi}p^{\prime
2}+d_{u}s^{2}\right)}\,,$ (79)

where $d_{\chi}=12-D_{\chi}^{\text{IR}}\in\mathbb{Z}$ and
$d_{u}=300-D_{u}^{\text{IR}}\in\mathbb{Z}$. If there exists a particular
solution $(m,p^{\prime},s)$ of the consistency conditions (14) such that no
integers $d_{\chi},d_{u}$ exists such that

$\displaystyle\frac{50}{3}m^{2}+d_{\chi}p^{2}+d_{u}s^{2}\in 4\mathbb{Z},$ (80)

then we conclude that composite fermions cannot match the
$\mathbb{Z}_{4}^{d\chi}[\text{CFU}]$ anomaly.

Consider $(m,p,s)=(1,0,2/3)$. This is a solution to the consistency conditions
and, therefore, corresponds to a CFU flux. In the presence of this CFU
background, the LHS of (80) becomes

$\displaystyle\frac{50}{3}+d_{u}\frac{4}{9}=\frac{150+4d_{u}}{9}\,.$ (81)

However, $150+4d_{u}\equiv 2\mod 4$ for any $d_{u}\in\mathbb{Z}$. Therefore,
we can conclude that for this theory, composite fermions cannot solely match
the $\mathbb{Z}_{4}^{d\chi}[\text{CFU}]$ anomaly in the IR.

### Matching by a condensate

Without composites, the anomalies are matched by spontaneous symmetry breaking
via condensates. First, the $2$-fermion condensate cannot match the anomalies,
as it breaks $SU(5)_{\chi}\times U(1)_{A}$ down to the anomalous subgroup
$SU(4)\times U(1)$. Next, consider the operator

$\displaystyle{\cal C}^{(ij)}=\psi^{2}\chi^{(i}\chi^{j)}\,,$ (82)

where $i,j$ are $SU(5)_{\chi}$ flavor indices, and in particular, this
condensate is in the two-index symmetric irrep of $SU(5)_{\chi}$. Thus, the
condensation of this operator breaks $SU(5)$ to the anomaly-free subgroup
$SO(5)$.

Under the action of $U(1)_{A}$, the condensate transforms as ${\cal
C}^{(ij)}=\psi^{2}\chi^{(i}\chi^{j)}\longrightarrow{\cal
C}^{\prime(ij)}=e^{2\pi(6\beta)}\psi^{2}\chi^{(i}\chi^{j)}$ where
$\beta\in[0,1)$ is the $U(1)_{A}$ parameter. So it appears that the condensate
is invariant under a discrete $\mathbb{Z}_{6}$ subgroup of $U(1)_{A}$. But
recall that the global symmetry group includes a division by the
$\mathbb{Z}_{3}$ center of the color group. $\mathbb{Z}_{3}$ is not a subgroup
of $SU(5)$, therefore it can only quotient $U(1)_{A}$, so the parameter
$\beta$ is in fact a $U(1)_{A}/\mathbb{Z}_{3}$ parameter and
$\beta\in[0,1/3)$. Therefore the condensate only exhibits an unbroken
$\mathbb{Z}_{2}$ symmetry, which has no global anomaly, and the $U(1)_{A}$
breaks to a non-anomalous subgroup.

The condensate also breaks $\mathbb{Z}_{4}^{d\chi}$ down to $\mathbb{Z}_{2}$,
leading to $2$ vacua connected via a domain wall. Recalling that the
$\mathbb{Z}_{4}^{d\chi}[\text{grav}]$ anomaly is only a $\mathbb{Z}_{2}$
phase, the unbroken subgroup $\mathbb{Z}_{2}\subset\mathbb{Z}_{4}^{d\chi}$ is
anomaly free (remember that $\mathbb{Z}_{2}$ is also free from nonperturbative
anomalies). In addition, the $\mathbb{Z}_{4}^{d\chi}[\mbox{CFU}]$ anomaly is
valued in $\mathbb{Z}_{2}$, meaning that the same condensate saturates it. The
breaking of $\mathbb{Z}_{4}^{d\chi}$ down to $\mathbb{Z}_{2}$ will also
automatically match the $[\mathbb{Z}_{4}^{d\chi}]^{3}$ anomaly, since
$\mathbb{Z}_{2}$ is anomaly free.

Let us examine the fate of the $U(1)_{A}[\mbox{CFU}]$ anomaly. First, when we
turn on the flavor center flux, the breaking of $SU(5)$ into $SO(5)$ matches
the anomaly, as the breaking causes the center of $SU(5)$ to break. Next, we
solely turn on the color and $U(1)_{A}$ fluxes. In this case,
$s\in\mathbb{Z}_{3}$, and the breaking of $U(1)_{A}$ down to $\mathbb{Z}_{2}$
implies that we are after $\mathbb{Z}_{2}[\mathbb{Z}_{3}]^{2}$ anomaly. The
anomaly coefficient can be read from Table 3, and according to (32), the
anomaly is automatically matched since $\mbox{gcd}(3,2)=1$.

We conclude that the global symmetry $G^{\scriptsize\mbox{g}}$ breaks down to
$SO(5)\times\frac{(\mathbb{Z}_{2}\subset
U(1)_{A})\times(\mathbb{Z}_{2}\subset\mathbb{Z}_{4}^{d\chi})}{\mathbb{Z}_{2}}$.
The first $\mathbb{Z}_{2}$ symmetry acts only on $\psi$, while the second
$\mathbb{Z}_{2}$ acts only on $\chi$. Then, from (75), we see that the
combination of these symmetries acts like the fermion number, which is left
intact in the IR. The extra modding by $\mathbb{Z}_{2}$ is employed to avoid
overcounting.

Since $SO(5)$ is the largest anomaly-free subgroup of $SU(5)$, this breaking
pattern minimizes the number of Goldstones and is the most favorable scenario.

###  4.3 $SU(6),k=1$

This theory has $2$ flavors of $\psi$ and 10 flavors of $\chi$, and thus, the
flavor symmetry is $SU(2)_{\psi}\times SU(10)_{\chi}$. The $U(1)_{A}$ charges
are

$\displaystyle q_{\psi}=-5\,,\quad q_{\chi}=2\,.$ (83)

Because $r=\mbox{gcd}(n_{\chi}T_{\chi},n_{\psi}T_{\psi})=\mbox{gcd}(40,16)=8$,
we may be tempted to conclude the theory has a $\mathbb{Z}_{8}$ chiral
symmetry. However, one can show that a $\mathbb{Z}_{2}$ subgroup of the
$\mathbb{Z}_{10}$ center of $SU(10)_{\chi}$ can be used to identify elements
of $\mathbb{Z}_{8}$:

$\displaystyle\chi$ $\displaystyle:$ $\displaystyle e^{-2\pi
i\frac{p^{\prime}}{10}}e^{2\pi i\frac{l}{8}}=e^{2\pi
i\frac{l^{\prime}}{8}}\,,$ (84)

for $l,l^{\prime}=1,2,...,7$. For example, setting $p^{\prime}=-5$ identifies
$\ell=1$ and $\ell=5$, etc. In addition, the solutions to the consistency
conditions (14) yield the division group
$\mathbb{Z}_{3}\times\mathbb{Z}_{2}\times\mathbb{Z}_{5}$. Thus, the faithful
global symmetry is

$\displaystyle G^{\scriptsize\mbox{g}}=\frac{SU(2)_{\psi}\times
SU(10)_{\chi}\times
U(1)_{A}}{\mathbb{Z}_{3}\times\mathbb{Z}_{2}\times\mathbb{Z}_{5}}\times\mathbb{Z}_{4}^{d\chi}\times\mathbb{Z}_{2}^{(1)}\,.$
(85)

The $\beta$-function indicates that the theory flows to an IR fixed point. At
$2$ loops, the coupling constant at the fixed point is
$\frac{g_{*}^{2}}{4\pi}\approx 0.094$. At $3$ loops, we obtain
$\frac{g_{*}^{2}}{4\pi}\approx 0.075$. Both values are much smaller than our
threshold value of $0.1$, and the $2$\- and $3$-loop analysis is only $10\%$
apart. Also, the $3$-loop to the $2$-loop ratio in (33) is $\approx 0.2$.
Thus, the fixed point is reliable. As we pointed out above, the lowest-order
bosonic operator in this theory, $F_{\mu\nu}\sigma^{\mu\nu}\chi\psi$,
necessitates the introduction of a color field to prevent its vanishing due to
statistics. This is a dimension-$5$ operator, and due to the smallness of the
coupling constant, we do not expect this operator to condense. Not to mention
that this operator by itself is not enough to match the full set of anomalies,
and higher-order condensates must also form to match them. We, thus, conclude
that the most probable scenario is that the theory flows to a CFT.

###  4.4 $SU(10),k=2$

The theory admits $3$ flavors of $\psi$ and $7$ flavors of $\chi$. The charges
of the fermions under $U(1)_{A}$ are

$\displaystyle q_{\psi}=-14\,,\quad q_{\chi}=9\,.$ (86)

We also have $r=\gcd(N_{\psi},N_{\chi})=4$, so that the theory is endowed with
a $\mathbb{Z}^{d\chi}_{4}$ chiral symmetry, which cannot be absorbed in a
combination of the centers of the color or flavor groups. After solving the
consistency equations, we obtain the faithful global symmetry group

$\displaystyle G^{\scriptsize\mbox{g}}=\frac{SU(3)_{\psi}\times
SU(7)_{\chi}\times
U(1)_{A}}{\mathbb{Z}_{5}\times\mathbb{Z}_{3}\times\mathbb{Z}_{7}}\times\mathbb{Z}^{d\chi}_{4}\times\mathbb{Z}^{(1)}_{2}\,.$
(87)

The theory develops a Banks-Zaks fixed point. The $2$ and $3$-loop values of
the coupling constant at the fixed point are
$\frac{g_{*}^{2}}{4\pi^{2}}\approx 0.059$ and
$\frac{g_{*}^{2}}{4\pi^{2}}\approx 0.046$, respectively. Also, the $3$-loop to
the $2$-loop ratio in (33) is $\approx 0.2$. Thus, like $SU(6),k=1$, this
theory is expected to flow to a CFT.

##  5 Bosonic theories

All gauge-invariant operators in this class of theories are bosonic. In the
following, we provide a systematic study of this class.

###  5.1 Conformal theories

We start by listing theories that flow to a conformal fixed point. These
theories are displayed in Table 4. In each case, the global symmetry is given
by

$\displaystyle G=\frac{SU(n_{\psi})\times SU(n_{\chi})\times
U(1)_{A}}{\Gamma}\times\mathbb{Z}_{p}^{d\chi}\times\mathbb{Z}_{2}^{(1)}\,.$
(88)

We also display the coupling constant $\frac{g_{*}^{2}}{4\pi^{2}}$ at the
$2$\- and $3$-loop fixed points. The smallness of the coupling constant and
its consistency between the $2$\- and $3$-loop calculations is an indicator of
the robustness of the fixed point. To quantify this robustness, we may
truncate the $\beta$-function to the second term in (33) and find the fixed
point is given by $\alpha_{*}=-\frac{4\pi\beta_{0}}{\beta_{1}}$. The existence
of such a fixed point implies that the first and second terms possess
comparable magnitudes. Consequently, the ratio between the third and second
(or first) term $\frac{\alpha_{*}\beta_{2}}{4\pi\beta_{1}}$ represents the
error incurred by neglecting the third term. A low ratio indicates the
perturbative nature of the fixed point.

The two theories $(N=20,k=4)$ and $(N=44,k=8)$ have the most reliable fixed
points. While the theory $(N=28,k=8)$ has
$\frac{\alpha_{*}\beta_{2}}{4\pi\beta_{1}}=1.12$, and its fixed point is under
question.

Theory | $n_{\psi}$ | $n_{\chi}$ | $\mathbb{Z}^{d\chi}_{p}$ | $\Gamma$ | $(q_{\psi},q_{\chi})$ | 2-loop | 3-loop | $\frac{\alpha_{*}\beta_{2}}{4\pi\beta_{1}}$
---|---|---|---|---|---|---|---|---
$SU(16),k=4$ | 3 | 5 | 2 | $\mathbb{Z}_{8}\times\mathbb{Z}_{3}\times\mathbb{Z}_{5}$ | $(-35,27)$ | 0.09 | $0.064$ | $0.62$
$SU(20),k=4$ | 4 | 6 | 2 | $\mathbb{Z}_{10}\times\mathbb{Z}_{4}\times\mathbb{Z}_{3}$ | $(-27,22)$ | $0.017$ | $0.015$ | $0.11$
$SU(28),k=8$ | 3 | 4 | 1 | $\mathbb{Z}_{7}\times\mathbb{Z}_{3}\times\mathbb{Z}_{4}$ | $(-52,45)$ | $0.086$ | $0.051$ | $1.12$
$SU(36),k=8$ | 4 | 5 | 1 | $\mathbb{Z}_{18}\times\mathbb{Z}_{4}\times\mathbb{Z}_{5}$ | $(-85,76)$ | $0.019$ | $0.016$ | $0.25$
$SU(44),k=8$ | 5 | 6 | 1 | $\mathbb{Z}_{11}\times\mathbb{Z}_{5}\times\mathbb{Z}_{6}$ | $(-126,115)$ | $0.0002$ | $0.0002$ | $0.003$
Table 4: A list of conformal bosonic theories.

###  5.2 Confining theories

####  5.2.1 $SU(8),k=4$

This theory was studied in Anber:2021iip . Here, we revisit it in light of the
discrete anomalies not discussed in Anber:2021iip . The theory admits
$n_{\psi}=1$ and $n_{\chi}=3$ flavors of fermions. The fermion charges under
$U(1)_{A}$ are

$\displaystyle q_{\psi}=-9\,,\quad q_{\chi}=5\,.$ (89)

Also, the theory admits a $\mathbb{Z}_{2}^{d\chi}$ discrete chiral symmetry.
Solving the consistency conditions (14) yield the faithful global symmetry

$\displaystyle G^{\scriptsize\mbox{g}}=\frac{SU(3)_{\chi}\times
U(1)_{A}}{\mathbb{Z}_{4}\times\mathbb{Z}_{3}}\times\mathbb{Z}_{2}^{d\chi}\times\mathbb{Z}_{2}^{(1)}\,.$
(90)

The theory admits many anomalies in Table 5.

Anomaly | Equation | Value
---|---|---
$\left[SU(3)_{\chi}\right]^{3}$ | $\mbox{dim}_{\chi}$ | 28
$U(1)_{A}\left[SU(3)_{\chi}\right]^{2}$ | $q_{\chi}\mbox{dim}_{\chi}$ | 140
$\mathbb{Z}_{2}^{d\chi}\left[SU(3)_{\chi}\right]^{2}$ | $\mbox{dim}_{\chi}$ | 28
$\mathbb{Z}_{2}^{d\chi}\left[U(1)_{A}\right]^{2}$ | $\kappa_{zu^{2}}=q_{\chi}^{2}\mbox{dim}_{\chi}n_{\chi}$ | 2100
$U(1)_{A}[\mbox{grav}]$ | $2(q_{\psi}\mbox{dim}_{\psi}+q_{\chi}\mbox{dim}_{\chi}n_{\chi})$ | 192
$\left[U(1)_{A}\right]^{3}$ | $\kappa_{u^{3}}=q_{\chi}^{3}\mbox{dim}_{\chi}n_{\chi}+q_{\psi}^{3}\mbox{dim}_{\psi}$ | -15744
$\mathbb{Z}_{2}^{d\chi}[\mbox{grav}]$ | $2\mbox{dim}_{\chi}n_{\chi}$ | 168 (trivial)
$U(1)_{A}[\mbox{CFU}]$ | $q_{\chi}\dim\chi\,Q_{\chi}+\kappa_{u^{3}}Q_{u}$ |  $\frac{280p^{\prime 2}}{3}-15744s^{2}$, $p^{\prime}\in\mathbb{Z}_{3}\,,s\in\mathbb{Z}_{12}$
$\mathbb{Z}_{2}^{d\chi}[\mbox{CFU}]$ | $n_{\chi}T_{\chi}Q_{c}+\dim\chi\,Q_{\chi}+\kappa_{zu^{2}}Q_{u}$ | $\frac{27m^{2}}{2}+\frac{56p^{\prime 2}}{3}+2100s^{2}\,,m\in\mathbb{Z}_{4}$
Table 5: Anomalies of $SU(8),k=4$.

In addition, the theory admits a $\mathbb{Z}_{2}^{d\chi}[\mbox{CFU}]$ anomaly,
which yields a phase of $\pi$ upon turning on a flux with, e.g.,
$(m,p^{\prime},s)=(1,0,\frac{1}{4})$, i.e., this is a $\mathbb{Z}_{4}\subset
U(1)_{A}$ flux. We also find that there is an anomaly of
$\mathbb{Z}_{2}^{d\chi}$ on a nonspin manifold, as the partition function
acquires a phase of $\pi$ by turning on a pure $\mathbb{Z}_{2}^{(1)}$ flux on
$\mathbb{CP}^{2}$.

In Anber:2021iip , it was argued that all the anomalies could be matched by
condensing two operators:

$\displaystyle{\cal C}_{1}^{i}=\psi\chi^{i}\,,\quad{\cal
C}_{2}^{i_{4}}=\epsilon_{i_{1}i_{2}i_{3}}\chi^{i_{1}}\chi^{i_{2}}\chi^{i_{3}}\chi^{i_{4}}\,.$
(91)

Let us review the anomaly matching using these two operators and comment on
why they cannot match the discrete anomalies.

Both operators ${\cal C}_{1}^{i}$ and ${\cal C}_{2}^{i}$ transform in the
defining representation of $SU(3)$ and break it down to the anomaly-free
$SU(2)$ (it has no Witten anomalies because the dimensions of the
representations are even). Yet, the condensation of ${\cal C}_{1}^{i}$ or
${\cal C}_{2}^{i}$ leaves behind an unbroken $SU(3)$ generator. We take ${\cal
C}_{1}^{i}\propto\delta_{i,1}$ and ${\cal C}_{2}^{i}\propto\delta_{i,1}$ and
parametrize the $SU(3)$ matrix that corresponds to the unbroken Cartan
generator of $SU(3)$ as
$\mbox{diag}\left(e^{i4\pi\alpha},e^{-i2\pi\alpha},e^{-i2\pi\alpha}\right)$.
Then, under $SU(3)_{\chi}\times U(1)_{A}\times\mathbb{Z}_{2}^{d\chi}$, the
operators transform as

$\displaystyle{\cal C}_{1}^{i}\longrightarrow
e^{i4\pi\alpha-i8\pi\beta+in\pi}{\cal C}_{1}^{i}\,,\quad{\cal
C}_{2}^{i}\longrightarrow e^{i40\pi\beta+i4\pi\alpha}{\cal C}_{2}^{i}\,,$ (92)

where $\beta$ corresponds to the $U(1)_{A}$ transformation, whereas $n=1$
corresponds to the $\mathbb{Z}_{2}^{d\chi}$ transformation. Taking
$\alpha=-\frac{5}{24}$, $\beta=\frac{1}{48}$, and $n=1$ leaves ${\cal
C}_{1}^{i}$ and ${\cal C}_{2}^{i}$ invariant under the combined
transformations of $SU(3)_{\chi}\times U(1)_{A}\times\mathbb{Z}_{2}^{d\chi}$.
This superficially hints at an unbroken $\mathbb{Z}_{24}$ symmetry. However,
owing to the modding by $\mathbb{Z}_{4}$ in (90), the genuine unbroken
subgroup is $\mathbb{Z}_{6}$. This unbroken symmetry can be written as
$\mathbb{Z}_{6}=\mathbb{Z}_{2}\times\mathbb{Z}_{3}$, where $\mathbb{Z}_{3}$ is
a genuine subgroup of $U(1)_{A}$. This can be seen by setting $n=0$, then we
find that ${\cal C}_{1}^{i}$ and ${\cal C}_{2}^{i}$ are left invariant by
taking $\alpha=-\frac{5}{12}$ and $\beta=\frac{1}{24}$. Remembering the
modding by $\mathbb{Z}_{4}$ in (90), we conclude that there is a
$\mathbb{Z}_{3}$ unbroken subgroup of $U(1)_{A}$.

It is straightforward to calculate the $\mathbb{Z}_{3}$ anomaly using (29) to
find that it is non-vanishing, meaning that the condensation of ${\cal
C}_{1}^{i}$ and ${\cal C}_{2}^{i}$ is not enough to match the complete set of
anomalies. The way out is to consider the condensation of the operator

$\displaystyle{\cal C}_{3}^{(ij)}=\psi^{2}\chi^{(i}\chi^{j)}\,,$ (93)

which transforms in the $2$-index symmetric representation of $SU(3)_{\chi}$
and breaks it down to $SO(3)$. $U(1)_{A}$ is broken to $\mathbb{Z}_{2}$, after
taking into account the modding by $\mathbb{Z}_{4}$ in (90). The
$\mathbb{Z}_{2}^{d\chi}[\mbox{CFU}]$ anomaly is automatically matched as
$U(1)_{A}$ is broken down to $\mathbb{Z}_{2}$ (remember, however, that this
$\mathbb{Z}_{2}$ is the fermion number since both fermions carry odd charges
under $U(1)_{A}$, and the fermion number is gauged). Recalling that we had to
turn on a $\mathbb{Z}_{4}\subset U(1)_{A}$ flux in the first place to see this
anomaly (a $\pi$ phase), the breaking of $U(1)_{A}$ to a smaller subgroup than
$\mathbb{Z}_{4}$ (in this case $\mathbb{Z}_{2}\subset\mathbb{Z}_{4}$)
trivializes the anomaly. Thus, at this level, one does not need to break
$\mathbb{Z}_{2}^{d\chi}$. This differs from the findings in Anber:2021iip ,
where it was argued that the CFU anomaly is not trivial. Here, we arrive at a
different IR condensate by scrutinizing the discrete subgroups of $U(1)_{A}$.

What about matching the anomaly of $\mathbb{Z}_{2}^{d\chi}$ on
$\mathbb{CP}^{2}$? Since this anomaly is valued in $\mathbb{Z}_{2}$, it can be
matched by a TQFT on a nonspin manifold, as was argued in Cordova:2019bsd .
Yet, another scenario is to form the condensate ${\cal C}_{1}^{i}$, which
breaks $\mathbb{Z}_{2}^{d\chi}$ to unity (remember that the
$\mathbb{Z}_{2}^{(1)}$ $1$-form symmetry is unbroken assuming confinement).
Thus, the condensation of both ${\cal C}_{1}^{i}$ and ${\cal C}_{3}^{(ij)}$
match all anomalies and break the global group down to $SO(3)$, resulting in
$2$ vacua (because of the breaking of $\mathbb{Z}_{2}^{d\chi}$) connected via
domain walls.

####  5.2.2 $SU(8),k=2$

This case was also considered briefly in Anber:2021iip . The theory admits $2$
flavors of $\psi$ fermions and $6$ flavors of $\chi$ fermions. The $U(1)_{A}$
charges of the fermions are

$\displaystyle q_{\psi}=-9\,,\quad q_{\chi}=5\,.$ (94)

Since $\mbox{gcd}(N_{\psi},N_{\chi})=4$, one may naively conclude that the
discrete symmetry is $\mathbb{Z}_{4}$. Yet, two elements of $\mathbb{Z}_{4}$
are identified with elements in $\mathbb{Z}_{2}\subset\mathbb{Z}_{6}$, where
$\mathbb{Z}_{6}$ is the center of $SU(6)_{\chi}$. This leaves us with
$\mathbb{Z}_{2}^{d\chi}$ as the genuine discrete group, which we take to act
solely on $\chi$. The faithful global symmetry is

$\displaystyle G^{\scriptsize\mbox{g}}=\frac{SU(2)_{\psi}\times
SU(6)_{\chi}\times
U(1)_{A}}{\mathbb{Z}_{4}\times\mathbb{Z}_{6}}\times\mathbb{Z}_{2}^{d\chi}\times\mathbb{Z}_{2}^{(1)}\,.$
(95)

The UV theory has the ‘t Hooft anomalies in Table 6.

Anomaly | Equation | Value
---|---|---
$[SU(6)_{\chi}]^{3}$ | $\text{dim}_{\chi}$ | $28$
$U(1)_{A}[\text{grav}]$ | $2(q_{\psi}\text{dim}_{\psi}+q_{\chi}\text{dim}_{\chi}n_{\chi})$ | $384$
$\mathbb{Z}_{2}^{d\chi}[\text{grav}]$ | $\text{dim}_{\chi}n_{\chi}$ |  $336$ (trivial)
$U(1)_{A}[SU(6)_{\chi}]^{2}$ | $q_{\chi}\text{dim}_{\chi}$ | $140$
$U(1)_{A}[SU(2)_{\psi}]^{2}$ | $q_{\psi}\text{dim}_{\psi}$ | $-324$
$[U(1)_{A}]^{3}$ | $q_{\psi}^{3}\text{dim}_{\psi}+q_{\chi}^{3}\text{dim}_{\chi}n_{\chi}$ | $-31488$
$\mathbb{Z}_{2}^{d\chi}[SU(6)_{\chi}]^{2}$ | $\mbox{dim}_{\chi}$ |  $28$ (trivial)
$\mathbb{Z}_{2}^{d\chi}[U(1)_{A}]^{2}$ | $q_{\psi}^{2}\mbox{dim}_{\psi}n_{\psi}+q_{\chi}^{2}\mbox{dim}_{\chi}n_{\chi}$ |  $4200$ (trivial)
Table 6: Anomalies of $SU(8),k=2$.

The $\mathbb{Z}_{2}^{d\chi}[\mbox{CFU}]$ anomaly does not provide new
information. However, there is a non-trivial
$\mathbb{Z}_{2}^{d\chi}[\mbox{CFU}]_{\mathbb{CP}^{2}}$ anomaly (a $\pi$ phase)
in the background of a CFU configuration with all fluxes turned on, e.g.,
$(m,p,p^{\prime},s)=(1,1,1,-5/12)$.

The condensation of the operator

$\displaystyle{\cal C}_{1\;j}^{i}=\psi_{j}\chi^{i}$ (96)

break $SU(2)_{\psi}\times SU(6)_{\chi}$ down to $SU(2)\times SU(4)$. The
unbroken $SU(4)$ is anomalous.

Another operator is

$\displaystyle{\cal
C}_{2}^{[i_{1}i_{2}]}=\psi^{2}\chi^{[i_{1}}\chi^{i_{2}]}\,,$ (97)

which is neutral under $SU(2)_{\psi}\times\mathbb{Z}_{2}^{d\chi}$ but
transforms in the $2$-index anti-symmetric representation of $SU(6)$ and
breaks it down to the anomaly-free $Sp(6)$121212Alternatively, one could
propose the formation of a condensate transforming in the $2$-index symmetric
representation of $SU(6)$. This condensate, however, would break $SU(6)$ down
to $SO(6)$, resulting in a larger number of Goldstones.. In addition, the
condensation of ${\cal C}_{2}^{[i_{1}i_{2}]}$ breaks $U(1)_{A}$ to the
anomaly-free $\mathbb{Z}_{2}$, after taking into account the modding by
$\mathbb{Z}_{4}$ in (95). What about the
$\mathbb{Z}_{2}^{d\chi}[\mbox{CFU}]_{\mathbb{CP}^{2}}$ anomaly? Remember that
one needs to turn on a configuration with $U(1)_{A}$ flux in
$\mathbb{Z}_{12}$. Since $U(1)_{A}$ breaks down to $\mathbb{Z}_{2}$, the
anomaly trivializes. Recall that this $\mathbb{Z}_{2}$ is the fermion number
since both fermions have odd charges under $U(1)_{A}$, and that the fermion
number is gauged. Thus, the condensation of ${\cal C}_{2}^{[i_{1}i_{2}]}$
leaves behind the unbroken $\frac{SU(2)_{\psi}\times
Sp(6)}{\mathbb{Z}_{2}}\times\mathbb{Z}_{2}^{d\chi}$ subgroup and matches all
anomalies131313The symplectic group $Sp(2N)$ has a $\mathbb{Z}_{2}$ center
symmetry, see, e.g., Anber:2014lba . This is why we needed to mod by
$\mathbb{Z}_{2}$ that is common between $Sp(6)$ and $SU(2)_{\psi}$..

####  5.2.3 $SU(12),k=4$

The number of flavors in this case is $n_{\psi}=2$ and $n_{\chi}=4$, and the
$U(1)_{A}$ charges are:

$\displaystyle q_{\psi}=-10\,,\quad q_{\chi}=7\,.$ (98)

Since $\mbox{gcd}(n_{\chi}T_{\chi},n_{\psi}T_{\psi})=\mbox{gcd}(40,28)=4$, one
may conclude that the theory is endowed with a $\mathbb{Z}_{4}$ chiral
symmetry that acts on $\chi$. However, this $\mathbb{Z}_{4}$ is the center of
the $SU(4)_{\chi}$ flavor symmetry. Therefore, the theory does not possess a
discrete chiral symmetry. Solving the consistency conditions (14), we find
that the faithful global symmetry group is:

$\displaystyle G^{\scriptsize\mbox{g}}=\frac{SU(2)_{\psi}\times
SU(4)_{\chi}\times
U(1)_{A}}{\mathbb{Z}_{3}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}}\times\mathbb{Z}^{(1)}_{2}\,.$
(99)

This theory is endowed with the anomalies in Table 7. The theory does not
possess a Witten anomaly of $SU(2)_{\psi}$ since $\mbox{dim}_{\psi}=66$ is an
even number.

Anomaly | Equation | Value
---|---|---
$[U(1)_{A}]^{3}$ | $q_{\chi}^{3}\mbox{dim}_{\chi}n_{\chi}+q_{\psi}^{3}n_{\psi}\mbox{dim}_{\psi}$ | -65448
$U(1)_{A}[SU(2)_{\psi}]^{2}$ | $q_{\psi}\mbox{dim}_{\psi}$ | -780
$U(1)_{A}[SU(4)_{\chi}]^{2}$ | $q_{\chi}\mbox{dim}_{\chi}$ | 462
$[SU(4)_{\chi}]^{3}$ | $\mbox{dim}_{\chi}$ | 66
$U(1)_{A}[\text{grav}]$ | $\kappa_{u^{3}}=q_{\chi}\mbox{dim}_{\chi}n_{\chi}+q_{\psi}n_{\psi}\mbox{dim}_{\psi}$ | 576
$U(1)_{A}[\text{CFU}]$ | $q_{\psi}\dim_{\psi}\,Q_{\psi}+q_{\chi}\dim_{\chi}\,Q_{\chi}+\kappa_{u^{3}}Q_{u}$ |  $390p^{2}+\frac{693}{2}p^{\prime 2}-65448s^{2}$, $p,p^{\prime}\in\mathbb{Z}_{2}$
Table 7: Anomalies of $SU(12),k=4$.

The $2$-loop and the $3$-loop $\beta$-functions predict fixed points at
$\frac{g_{*}^{2}}{4\pi}=0.514$ and $0.202$, respectively. Both values are
large for the fixed points to be robust.

In searching for candidates that break the symmetries spontaneously, let us
study the bilinear condensate:

$\displaystyle{\cal
C}_{j}^{i}=\epsilon^{a_{1}...a_{12}}\left(f_{\mu\nu}^{c}\right)_{a_{2}}^{a_{13}}\sigma^{\mu\nu}\epsilon_{\alpha_{1}\alpha_{2}}\psi^{\alpha_{1}}_{j,\,(a_{1}a_{13})}\chi^{\alpha_{2},\,i}_{[a_{3}..a_{12}]}\,,\quad
j=1,2\,,i=1,2,3,4,$ (100)

where, as usual, $a_{1},a_{2},..$ are color indices, $\alpha_{1},\alpha_{2}$
are spinor indices, while $j$ and $i$ are respectively $SU(2)_{\psi}$ and
$SU(4)_{\chi}$ indices. The transformation of ${\cal C}_{j}^{i}$ is noteworthy
as it occurs in the fundamental representation of $SU(2)\psi$ and the anti-
fundamental representation of $SU(4)\chi$. Consequently, upon condensation, it
has the potential to break down $SU(2)_{\psi}\times SU(4)_{\chi}$ to
$SU(2)_{V}\times SU(2)$. This symmetry-breaking pattern can be explained as
follows Li:1973mq .

To create an invariant potential for the $4\times 2$ matrix ${\cal
C}_{j}^{i}$, we define the $4\times 4$ matrix
$\Phi^{i,i^{\prime}}\equiv\sum_{j=1}^{2}{\cal C}_{j}^{i}{\cal
C}_{j}^{i^{\prime}}$ . By considering the effective potential as a trace over
quadratic and quartic terms of $\Phi^{i,i^{\prime}}$, we might initially
assume that we can transform to a basis where $\Phi^{i,i^{\prime}}$ becomes a
non-degenerate diagonal matrix. However, this assumption leads to a
contradiction because the $4\times 1$ column vectors in $\Phi^{i,i^{\prime}}$
are dependent due to the construction of $\Phi^{i,i^{\prime}}$ from a $4\times
2$ matrix. In other words, $\Phi^{i,i^{\prime}}$ possesses two zero
eigenvalues. Hence, we conclude that we can only transform to a basis that
diagonalizes $SU(2){\psi}\times(SU(2)\subset SU(4){\chi})$. This results in
the diagonal (vector-like) matrix $SU(2)_{V}$, while $SU(4-2)=SU(2)\subset
SU(4){\chi}$ remains unbroken. Both $SU(2)_{V}$ and $SU(2)\subset SU(4){\chi}$
are subgroups devoid of anomalies. The potential anomaly, namely the Witten
anomaly, does not afflict any of these subgroups. The UV particle content
ensures that the number of fermions transforming under $SU(2)\psi$ and
$SU(2)\subset SU(4)_{\chi}$ is $\mbox{dim}{\psi}=78$ and
$\mbox{dim}{\chi}=66$, respectively, both of which are even numbers.
Therefore, none of these groups can exhibit Witten anomalies.

Moreover, due to the $\mathbb{Z}_{3}$ modding in (99), the axial symmetry
$U(1)_{A}/\mathbb{Z}_{3}$ identifies a transformation phase $\alpha$ with
$\alpha+\frac{2\pi}{3}$. The charge of the condensate ${\cal C}_{j}^{i}$ under
$U(1)_{A}$ is $-3$, leading to the breaking of $U(1)_{A}$ to unity. Hence, we
conclude that the $2$-fermion condensate ${\cal C}_{j}^{i}$ successfully
saturates all the anomalies and breaks the global symmetry down to
$\frac{SU(2)_{V}\times(SU(2)\subset SU(4){\chi})}{\mathbb{Z}_{2}}$, where we
mod by the $\mathbb{Z}_{2}$ common center of both groups.

####  5.2.4 $SU(12),k=8$

The number of flavors in this case is $n_{\psi}=1$ and $n_{\chi}=2$ and the
$U(1)_{A}$ charges are:

$\displaystyle q_{\psi}=-10\,,\quad q_{\chi}=7\,.$ (101)

Given that
$r=\mbox{gcd}(n_{\psi}T_{\psi},n_{\chi}T_{\chi})=\mbox{gcd}(14,20)=2$, we may
conclude that the theory has a $\mathbb{Z}_{2}$ discrete chiral symmetry. Yet,
one can absorb this $\mathbb{Z}_{2}$ in the center of $SU(2)_{\chi}$, leaving
behind no genuine discrete symmetry. After solving the consistency conditions,
we find that the faithful global symmetry group is:

$\displaystyle G^{\scriptsize\mbox{g}}=\frac{SU(2)_{\chi}\times
U(1)_{A}}{\mathbb{Z}_{3}\times\mathbb{Z}_{2}}\times\mathbb{Z}^{(1)}_{2}\,.$
(102)

The theory possesses the anomalies in Table 8.

Anomaly | Equation | Value
---|---|---
$[U(1)_{A}]^{3}$ | $\kappa_{u^{3}}=q_{\chi}^{3}\text{dim}_{\chi}n_{\chi}+q_{\psi}^{3}\text{dim}_{\psi}$ | $-32724$
$U(1)_{A}[SU(2)_{\chi}]^{2}$ | $q_{\chi}\text{dim}_{\chi}$ | $462$
$U(1)_{A}[\text{grav}]$ | $q_{\chi}\text{dim}_{\chi}n_{\chi}+q_{\psi}\text{dim}_{\psi}$ | $288$
$U(1)_{A}[\text{CFU}]$ | $q_{\chi}\dim_{\chi}\,Q_{\chi}+\kappa_{u^{3}}Q_{u}$ |  $231p^{\prime 2}-32724s^{2}$, $p^{\prime}\in\mathbb{Z}_{2}$
Table 8: Anomalies of $SU(12),k=8$.

The potential Witten anomaly of $SU(2)_{\chi}$ is absent because
$\mbox{dim}_{\chi}=66$ is an even number.

The $2$-loop and $3$-loop $\beta$-functions do not predict fixed points, and
the theory needs to break its symmetries spontaneously by forming condensates.
The operator

$\displaystyle{\cal C}_{1}^{i}=\psi\chi^{i}\,,$ (103)

where the index $i$ is the $SU(2)_{\chi}$ flavor, breaks the global symmetry
down to $U(1)$. To see that, let us fix the vacuum to be $\left[1\quad
0\right]^{T}$. Then, if a $U(1)$ generator is left unbroken by the vacuum, one
should find a nontrivial solution to

$\displaystyle\exp\left[i2\pi\beta\left[\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right]\right]e^{-i6\pi\alpha I_{2\times
2}}\left[\begin{array}[]{c}1\\\
0\end{array}\right]=\left[\begin{array}[]{c}1\\\ 0\end{array}\right]\,.$ (110)

It is easy to check that the solution $\beta=3\alpha$ satisfies the above
equation, which is the unbroken $U(1)$ direction. The unbroken $U(1)$ symmetry
inherits the UV mixed $U(1)_{A}[\mbox{grav}]$ anomaly, and thus, condensing
${\cal C}_{1}^{i}$ is not enough to match the anomalies.

Another operator that can condense is

$\displaystyle{\cal C}_{2}=\epsilon_{ij}\psi\psi\chi^{i}\chi^{j}\,,$ (111)

with possible insertions of gluon fields. The operator ${\cal C}_{2}$ is
singlet under $SU(2)$, but it has a charge $-6$ under $U(1)_{A}$. Because of
the modding by $\mathbb{Z}_{3}$ in (102), the condensation of ${\cal C}_{2}$
breaks $U(1)_{A}$ down to $\mathbb{Z}_{2}$, an anomaly-free subgroup. We
conclude that the condensation of ${\cal C}_{2}$ is enough to match the
anomalies, a scenario with the minimum number of Goldstones.

####  5.2.5  $SU(20),k=8$

The number of flavors is $n_{\psi}=2$ and $n_{\chi}=3$, while the $U(1)_{A}$
charges are:

$\displaystyle q_{\psi}=-27\,,\quad q_{\chi}=22\,.$ (112)

Since $r=\mbox{gcd}(n_{\psi}T_{\psi},n_{\chi}T_{\chi})=(36,66)=2$, we might
conclude that the theory possesses a $\mathbb{Z}_{2}$ chiral symmetry.
However, this symmetry can be rotated away in the following way. First,
according to our choice, the would-be chiral symmetry acts only on $\chi$.
Thus, $(\psi,\chi)\longrightarrow(\psi,-\chi)$ under this $\mathbb{Z}_{2}$.
Next, we apply a transformation by $(-1)^{F}$, which sends
$(\psi,-\chi)\longrightarrow(-\psi,\chi)$. Finally, we apply another
transformation by the center of $SU(2)_{\psi}$, which sends
$(-\psi,\chi)\longrightarrow(\psi,\chi)$. This shows that the theory does not
possess a discrete chiral symmetry. Finding the solutions to the consistency
conditions, the faithful global symmetry group is:

$\displaystyle G^{\scriptsize\mbox{g}}=\frac{SU(2)_{\psi}\times
SU(3)_{\chi}\times
U(1)_{A}}{\mathbb{Z}_{5}\times\mathbb{Z}_{2}\times\mathbb{Z}_{3}}\times\mathbb{Z}^{(1)}_{2}\,.$
(113)

The anomalies of this theory are given in Table 9.

Anomaly | Equation | Value
---|---|---
$\left[U(1)_{A}\right]^{3}$ | $\kappa_{u^{3}}=q_{\chi}^{3}\text{dim}_{\chi}n_{\chi}+q_{\psi}^{3}\text{dim}_{\psi}$ | $-2197500$
$U(1)_{A}\left[SU(2)_{\psi}\right]^{2}$ | $q_{\psi}n_{\psi}\text{dim}_{\psi}$ | $-5670$
$U(1)_{A}\left[SU(3)_{\chi}\right]^{2}$ | $q_{\chi}n_{\chi}\text{dim}_{\chi}$ | $4180$
$\left[SU(3)_{\chi}\right]^{3}$ | $\text{dim}_{\chi}$ | $190$
$U(1)_{A}\left[\text{grav}\right]$ | $2(q_{\chi}\text{dim}_{\chi}n_{\chi}+q_{\psi}\text{dim}_{\psi})$ | $2400$
$U(1)_{A}[\text{CFU}]$ | $q_{\psi}\dim_{\psi}\,Q_{\psi}+q_{\chi}\dim_{\chi}\,Q_{\chi}+\kappa_{u^{3}}Q_{u}$ | $-2835p^{2}+\frac{8360}{3}p^{\prime 2}-2197500s^{2}$
Table 9: Anomalies of $SU(20),k=8$.

Since $SU(2)_{\psi}$ is an anomaly-free group, it does not need to break. The
scenario that gives the lowest number of Goldstones amounts to breaking
$SU(3)\chi\times U(1)_{A}$ to an anomaly-free subgroup. This can be achieved
by condensing

$\displaystyle{\cal C}^{(ij)}=\psi^{2}\chi^{(i}\chi^{j)}\,,$ (114)

which is singlet under $SU(2)_{\psi}$ and transforms in the $2$-index
symmetric representation of $SU(3)$ breaking it to $SO(3)$. As before, this
condensate also breaks $U(1)_{A}$ to the anomaly-free subgroup
$\mathbb{Z}_{2}$. Thus, the IR unbroken $0$-form symmetry is
$\frac{SU(2)\psi\times(\mathbb{Z}_{2}\subset U(1)_{A})}{\mathbb{Z}_{2}}\times
SO(3)$.

##  6 Summary

Theory | Global Symmetries | Condensate(s) | IR Symmetries
---|---|---|---
$SU(5),k=1$ | $\frac{SU(9)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{5}\times\mathbb{Z}_{9}}$ | $\psi\chi^{9}\chi^{(i}\chi^{j)}$ | $SO(9)\times(\mathbb{Z}_{10}\subset U(1)_{A})$
$SU(6),k=1$ | $\frac{SU(2)_{\psi}\times SU(10)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{3}\times\mathbb{Z}_{2}\times\mathbb{Z}_{5}}\times\mathbb{Z}_{4}^{d\chi}$ | — | CFT
$SU(6),k=2$ | $\frac{SU(5)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{3}\times\mathbb{Z}_{5}}\times\mathbb{Z}_{4}^{d\chi}$ | $\psi^{2}\chi^{(i}\chi^{j)}$ |  $SO(5)\times$ $\frac{(\mathbb{Z}_{2}\subset U(1)_{A})\times(\mathbb{Z}_{2}\subset\mathbb{Z}_{4}^{d\chi})}{\mathbb{Z}_{2}}$
$SU(10),k=2$ | $\frac{SU(3)_{\psi}\times SU(7)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{5}\times\mathbb{Z}_{3}\times\mathbb{Z}_{7}}\times\mathbb{Z}^{d\chi}_{4}$ | — | CFT
$SU(8),k=2$ | $\frac{SU(2)_{\psi}\times SU(6)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{4}\times\mathbb{Z}_{6}}\times\mathbb{Z}_{2}^{d\chi}$ | $\psi^{2}\chi^{[i}\chi^{j]}$ | $\frac{SU(2)_{\psi}\times Sp(6)}{\mathbb{Z}_{2}}\times\mathbb{Z}_{2}^{d\chi}$
$SU(8),k=4$ | $\frac{SU(3)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{4}\times\mathbb{Z}_{3}}\times\mathbb{Z}_{2}^{d\chi}$ | $\psi\chi^{i},\psi^{2}\chi^{(i}\chi^{j)}$ | $SO(3)$
$SU(12),k=4$ | $\frac{SU(2)_{\psi}\times SU(4)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{6}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}}$ | $\psi_{i}\chi^{j}$ | $\frac{SU(2)_{V}\times(SU(2)\subset SU(4){\chi})}{\mathbb{Z}_{2}}$
$SU(12),k=8$ | $\frac{SU(2)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{3}\times\mathbb{Z}_{2}}$ | $\epsilon_{ij}\psi^{2}\chi^{i}\chi^{j}$ | $\frac{SU(2)_{\chi}\times(\mathbb{Z}_{2}\subset U(1)_{A})}{\mathbb{Z}_{2}}$
$SU(16),k=4$ | $\frac{SU(3)_{\psi}\times SU(5)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{8}\times\mathbb{Z}_{3}\times\mathbb{Z}_{5}}\times\mathbb{Z}^{d\chi}_{2}$ | — | CFT
$SU(20),k=8$ | $\frac{SU(2)_{\psi}\times SU(3)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{5}\times\mathbb{Z}_{2}\times\mathbb{Z}_{3}}$ | $\psi^{2}\chi^{(i}\chi^{j)}$ | $\frac{SU(2)_{\psi}\times(\mathbb{Z}_{2}\subset U(1)_{A})}{\mathbb{Z}_{2}}\times SO(3)$
$SU(20),k=4$ | $\frac{SU(4)_{\psi}\times SU(6)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{10}\times\mathbb{Z}_{4}\times\mathbb{Z}_{3}}\times\mathbb{Z}^{d\chi}_{2}$ | — | CFT
$SU(28),k=8$ | $\frac{SU(3)_{\psi}\times SU(4)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{7}\times\mathbb{Z}_{3}\times\mathbb{Z}_{4}}$ | — | CFT
$SU(36),k=8$ | $\frac{SU(4)_{\psi}\times SU(5)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{18}}$ | — | CFT
$SU(44),k=8$ | $\frac{SU(5)_{\psi}\times SU(6)_{\chi}\times U(1)_{A}}{\mathbb{Z}_{11}\times\mathbb{Z}_{5}\times\mathbb{Z}_{6}}$ | — | CFT
Table 10: A summary of the $2$-index chiral theories, their global symmetries,
and their IR realizations. Theories with $N$ even also enjoy a
$\mathbb{Z}_{2}^{(1)}$ $1$-form symmetry acting on the Wilson lines. This
symmetry is assumed to be unbroken in theories that confine.

In this paper, we exhaustively scrutinized the $2$-index chiral gauge
theories. By studying the $2$-loop and $3$-loop $\beta$-functions, we could
pinpoint a few theories that may flow to an IR CFT. Theories that do not admit
a fixed point break its global symmetries. We considered scenarios that give
the minimal number of IR Goldstones, as this lowers the free energy of the
theory. We paid particular attention to the anomaly-matching conditions and
ensured that the condensates match any discrete subgroup of $U(1)_{A}$. Our
theories, their global symmetries, the proposed IR phase condensates, and the
unbroken IR symmetries are shown in Table 10. The first $4$ theories are
fermionic, while the rest are bosonic.

Our investigation included a closer examination of the CFU anomalies one of
the authors studied in the previous work Anber:2021iip , giving a better
interpretation of this class of anomalies in the light of the discrete-anomaly
matching conditions. The general finding is that matching the full set of
anomalies and, in particular, the anomalies of the discrete subgroups of the
axial $U(1)_{A}$ symmetry necessitates the formation of multiple higher-order
condensates. One expects such higher-order condensates to form in strongly-
coupled theories. Here, their formation is explained via the constraints of
the anomaly-matching conditions. We also employed a systematic approach to
search for massless composite fermions that could match the anomalies in the
case of fermionic theories. We were not able to find such composites. In one
case, we used the CFU anomaly to show that a set of composites cannot solely
match this anomaly, hinting at a deeper reason why the composites could not be
found.

Our work provides a systematic approach that can be applied to study other
classes of strongly-coupled phenomena, including different chiral gauge
theories.

Acknowledgments:

We would like to thank Nakarin Lohitsiri and Erich Poppitz for various
illuminating discussions and comments on the manuscript. M.A. acknowledges the
hospitality at the University of Toronto, where part of this work was
completed. This work is supported by STFC through grant ST/T000708/1.

## Appendix

##  Appendix A Obtaining the discrete chiral symmetry

In this appendix, we show that there is a discrete symmetry $\mathbb{Z}_{r}$,
where $r=\mbox{gcd}(N_{\psi},N_{\chi})$, that acts on $\chi$. To this end, we
conisder the groups $\mathbb{Z}_{N_{\psi}p_{\psi}+N_{\chi}p_{\chi}}$ and
$U(1)_{A}$ we discussed in the text. Under
$U(1)_{A}\times\mathbb{Z}_{N_{\psi}p_{\psi}+N_{\chi}p_{\chi}}$, $\psi$
transforms as

$\displaystyle\psi\longrightarrow e^{2\pi iaq_{\psi}\alpha}e^{2\pi
ip_{\psi}\frac{l}{N_{\psi}p_{\psi}+N_{\chi}p_{\chi}}}\psi\,,$ (115)

where $l\in\mathbb{Z}_{N_{\psi}p_{\psi}+N_{\chi}p_{\chi}}$, $a$ is a charge
factor, and $\alpha\in[0,1)$. This transformation leaves $\psi$ invariant if

$\displaystyle
aq_{\psi}\alpha+p_{\psi}\frac{l}{N_{\psi}p_{\psi}+N_{\chi}p_{\chi}}=k_{1}\in\mathbb{Z}\implies\alpha=\frac{k_{1}}{aq_{\psi}}-\frac{p_{\psi}l}{aq_{\psi}(N_{\psi}p_{\psi}+N_{\chi}p_{\chi})}\,.$
(116)

Note that $k_{1}$ can be freely chosen. Then, $\chi$ transforms under
$U(1)_{A}\times\mathbb{Z}_{N_{\psi}p_{\psi}+N_{\chi}p_{\chi}}$ as:

$\displaystyle\chi$ $\displaystyle\longrightarrow e^{2\pi
iaq_{\chi}\alpha}e^{2\pi
ip_{\chi}\frac{l}{N_{\psi}p_{\psi}+N_{\chi}p_{\chi}}}\chi=e^{2\pi
i\left(aq_{\chi}\left(\frac{k_{1}}{aq_{\psi}}-\frac{p_{\psi}l}{aq_{\psi}(N_{\psi}p_{\psi}+N_{\chi}p_{\chi})}\right)+p_{\chi}\frac{l}{N_{\psi}p_{\psi}+N_{\chi}p_{\chi}}\right)}\chi$
(117) $\displaystyle=e^{2\pi
i\left(\frac{q_{\chi}}{q_{\psi}}k_{1}+\frac{l}{N_{\psi}p_{\psi}+N_{\chi}p_{\chi}}\left(p_{\chi}-p_{\psi}\frac{q_{\chi}}{q_{\psi}}\right)\right)}\chi=e^{2\pi
i\left(\frac{q_{\chi}}{q_{\psi}}k_{1}+\frac{l}{(N_{\psi}p_{\psi}+N_{\chi}p_{\chi})q_{\psi}}\left(-p_{\chi}\frac{N_{\chi}}{r}-p_{\psi}\frac{N_{\psi}}{r}\right)\right)}\chi\,,$
$\displaystyle=e^{2\pi
i\left(\frac{q_{\chi}}{q_{\psi}}k_{1}-\frac{l}{rq_{\psi}}\right)}\chi\,,$

where $r=\gcd(N_{\psi},N_{\chi})$ and we used $q_{\psi}=-\frac{N_{\chi}}{r}$
and $q_{\chi}=\frac{N_{\psi}}{r}$. We can rewrite $l=m_{1}+m_{2}r$, where
$m_{1}=0,1,\dots,r-1$ and $m_{2}\in\mathbb{Z}$. Bezout’s theorem also tells us
that since $r=\gcd(N_{\psi},N_{\chi})$, there are integers $k_{1},k_{2}$ such
that $m_{2}r=k_{1}N_{\psi}+k_{2}N_{\chi}$. Applying this to the transformation
of $\chi$ gives us:

$\displaystyle\chi$ $\displaystyle\longrightarrow e^{2\pi
i\left(\frac{q_{\chi}}{q_{\psi}}k_{1}-\frac{l}{rq_{\psi}}\right)}\chi=e^{2\pi
i\left(\frac{q_{\chi}}{q_{\psi}}k_{1}-\frac{m_{1}+m_{2}r}{rq_{\psi}}\right)}\chi=e^{2\pi
i\left(\frac{m_{1}+m_{2}r}{N_{\chi}}-\frac{N_{\psi}}{N_{\chi}}k_{1}\right)}\chi$
(118) $\displaystyle=e^{2\pi i\frac{m_{1}}{N_{\chi}}}e^{2\pi
i\frac{m_{2}r-N_{\psi}k_{1}}{N_{\chi}}}\chi=e^{2\pi
i\frac{m_{1}}{N_{\chi}}}e^{2\pi ik_{2}}\chi=e^{2\pi
i\frac{m_{1}}{N_{\chi}}}\chi\,.$

Since $m_{1}=0,1,\dots,r-1$, there are only $r$ distinct transformations
generated by $U(1)_{A}\times\mathbb{Z}_{N_{\psi}p_{\psi}+N_{\chi}p_{\chi}}$,
and the symmetry group that acts on $\chi$ is $\mathbb{Z}_{r}$. For our
purposes, we will assume that under $\mathbb{Z}_{r}$, $\chi$ transforms with
charge 1 (in principle, we could fix any charge). Finally, one needs to check
whether this $\mathbb{Z}_{r}$ is a genuine symmetry in the sense that it
cannot be absorbed in the center of color or flavor groups. This will be done
on a case-by-case basis.

##  Appendix B The 3-loop $\beta$-function and the IR fixed points

The 3-loop $\beta$ function is given by (see Caswell:1974gg ; Dietrich:2006cm
; Zoller:2016sgq )

$\displaystyle\begin{split}\beta(g)=&-\beta_{0}\frac{g^{3}}{(4\pi)^{2}}-\beta_{1}\frac{g^{5}}{(4\pi)^{4}}-\beta_{2}\frac{g^{7}}{(4\pi)^{6}}~{},\\\\[3.0pt]
\beta_{0}=&\frac{11}{6}C_{2}(G)-\sum_{\cal R}\frac{1}{3}T_{\cal R}n_{\cal
R}~{},\\\\[3.0pt] \beta_{1}=&\frac{34}{12}C_{2}^{2}(G)-\sum_{{\cal
R}}\left\\{\frac{5}{6}n_{\cal R}C_{2}(G)T_{\cal R}+\frac{n_{\cal
R}}{2}C_{2}({\cal R})T_{\cal R}\right\\}~{},\\\\[3.0pt]
\beta_{2}=&\frac{2857}{432}C_{2}^{3}(G)-\sum_{\cal{R}}\frac{n_{\cal R}T_{\cal
R}}{4}\left[-\frac{C_{2}^{2}({\cal R})}{2}+\frac{205C_{2}(G)C_{2}({\cal
R})}{36}+\frac{1415C_{2}^{2}(G)}{108}\right]\\\\[3.0pt]
&+\quad\sum_{\cal{R},\cal{R}^{\prime}}\frac{n_{\cal R}n_{\cal
R}^{\prime}T_{\cal R}T_{\cal R^{\prime}}}{16}\left[\frac{44C_{2}({\cal
R})}{18}+\frac{158C_{2}(G)}{54}\right]\,.\end{split}$ (119)

Here, $G$ denotes the adjoint representation, and $n_{\cal R}$ is the number
of the Weyl flavors in representation ${\cal R}$. Also, $C_{2}({\cal R})$ is
the quadratic Casimir operator of representation ${\cal R}$, defined as

$\displaystyle t^{a}_{\cal R}t^{a}_{\cal R}=C_{2}({\cal R})\mathbf{1}_{\cal
R}~{}.$ (120)

We reserve $C_{2}(G)$ for the quadratic Casimir of the adjoint representation.
$T_{\cal R}$ is the Dynkin index of ${\cal R}$, which is defined by

$\displaystyle\mbox{tr}\left[t^{a}_{\cal R}t^{b}_{\cal R}\right]=T_{\cal
R}\delta^{ab}~{}.$ (121)

From Eqs. (120) and (121), we easily obtain the useful relation

$\displaystyle T_{\cal R}\mbox{dim}_{G}=C_{2}({\cal R})\mbox{dim}_{\cal
R}~{},$ (122)

where $\mbox{dim}_{\cal R}$ is the dimension of $\cal R$.

In particular, we have $C_{2}(G)=2N$, $\mbox{dim}_{G}=N^{2}-1$,
$T_{\psi}=N+2$, $\mbox{dim}_{\psi}=\frac{N(N+1)}{2}$,
$C_{2}(\psi)=\frac{2(N+2)(N-1)}{N}$, $T_{\chi}=N-2$,
$\mbox{dim}_{\chi}=\frac{N(N-1)}{2}$, $C_{2}(\chi)=\frac{2(N-2)(N+1)}{N}$.
Then, the values of $\beta_{0}$ to $\beta_{2}$ are

$\displaystyle\begin{split}\beta_{0}=&\
\frac{1}{3}\left[11N-\frac{2}{k}(N^{2}-8)\right]~{},\\\\[3.0pt] \beta_{1}=&\
\frac{2\left(-48+76N^{2}+17kN^{3}-8N^{4}\right)}{3kN}~{},\\\\[3.0pt]
\beta_{2}=&\
\frac{1}{54k^{2}N^{2}}\Big{[}2857k^{2}N^{5}+N(-8448+12448N^{2}-2584N^{4}+145N^{6})\\\\[3.0pt]
&\quad-2k(864+3948N^{2}-8945N^{4}+988N^{6})\Big{]}~{}.\end{split}$ (123)

Assuming that $\beta_{0}>0$ and $\beta_{1}<0$, the theory develops an IR fixed
point to $2$-loops. The value of the coupling constant at the fixed point is

$\displaystyle\alpha_{*}\equiv\frac{g_{*}^{2}}{4\pi}=-\frac{4\pi\beta_{0}}{\beta_{1}}=\frac{2\pi
N\left(16+11kN-2N^{2}\right)}{48-76N^{2}-17kN^{3}+8N^{4}}~{}.$ (124)

To assess the stability of this fixed point, we can examine the roots of the
$\beta$-function when the 3-loop term is taken into account.

## References

  * (1) S. Bolognesi, K. Konishi, and A. Luzio, Dynamical Abelianization and anomalies in chiral gauge theories, JHEP 12 (2022) 110, [arXiv:2206.00538].
  * (2) S. Bolognesi, K. Konishi, and A. Luzio, Probing the dynamics of chiral $SU(N)$ gauge theories via generalized anomalies, Phys. Rev. D 103 (2021), no. 9 094016, [arXiv:2101.02601].
  * (3) S. Bolognesi, K. Konishi, and A. Luzio, Dynamics from symmetries in chiral $SU(N)$ gauge theories, JHEP 09 (2020) 001, [arXiv:2004.06639].
  * (4) S. Bolognesi, K. Konishi, and M. Shifman, Patterns of symmetry breaking in chiral QCD, Phys. Rev. D 97 (2018), no. 9 094007, [arXiv:1712.04814].
  * (5) Y. Bai and D. Stolarski, Phases of confining SU(5) chiral gauge theory with three generations, JHEP 03 (2022) 113, [arXiv:2111.11214].
  * (6) C. Csáki, H. Murayama, and O. Telem, Some exact results in chiral gauge theories, Phys. Rev. D 104 (2021), no. 6 065018, [arXiv:2104.10171].
  * (7) C. Csáki, H. Murayama, and O. Telem, More exact results on chiral gauge theories: The case of the symmetric tensor, Phys. Rev. D 105 (2022), no. 4 045007, [arXiv:2105.03444].
  * (8) M. M. Anber, S. Hong, and M. Son, New anomalies, TQFTs, and confinement in bosonic chiral gauge theories, JHEP 02 (2022) 062, [arXiv:2109.03245].
  * (9) P. B. Smith, A. Karasik, N. Lohitsiri, and D. Tong, On discrete anomalies in chiral gauge theories, JHEP 01 (2022) 112, [arXiv:2106.06402].
  * (10) C.-H. Sheu and M. Shifman, Consistency of chiral symmetry breaking in chiral Yang-Mills theory with adiabatic continuity, Phys. Rev. D 107 (2023), no. 5 054030, [arXiv:2212.14794].
  * (11) M. M. Anber and E. Poppitz, On the baryon-color-flavor (BCF) anomaly in vector-like theories, JHEP 11 (2019) 063, [arXiv:1909.09027].
  * (12) M. M. Anber and S. Baker, Natural inflation, strong dynamics, and the role of generalized anomalies, Phys. Rev. D 102 (2020), no. 10 103515, [arXiv:2008.05491].
  * (13) M. M. Anber, N. Lohitsiri, and T. Sulejmanpasic, Remarks on QCD4 with fundamental and adjoint matter, arXiv:2306.01849.
  * (14) M. M. Anber and E. Poppitz, Deconfinement on axion domain walls, JHEP 03 (2020) 124, [arXiv:2001.03631].
  * (15) T. Nakajima, T. Sakai, and R. Yokokura, BCF anomaly and higher-group structure in the low energy effective theories of mesons, JHEP 01 (2023) 175, [arXiv:2212.12987].
  * (16) M. M. Anber and E. Poppitz, Generalized ’t Hooft anomalies on non-spin manifolds, JHEP 04 (2020) 097, [arXiv:2002.02037].
  * (17) Y. Tanizaki, Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking, JHEP 08 (2018) 171, [arXiv:1807.07666].
  * (18) H. Shimizu and K. Yonekura, Anomaly constraints on deconfinement and chiral phase transition, Phys. Rev. D 97 (2018), no. 10 105011, [arXiv:1706.06104].
  * (19) X.-z. Dai and D. S. Freed, eta invariants and determinant lines, J. Math. Phys. 35 (1994) 5155–5194, [hep-th/9405012]. [Erratum: J.Math.Phys. 42, 2343–2344 (2001)].
  * (20) E. Witten and K. Yonekura, Anomaly Inflow and the $\eta$-Invariant, in The Shoucheng Zhang Memorial Workshop, 9, 2019.  arXiv:1909.08775.
  * (21) S. S. Razamat and D. Tong, Gapped Chiral Fermions, Phys. Rev. X 11 (2021), no. 1 011063, [arXiv:2009.05037].
  * (22) I. n. García-Etxebarria and M. Montero, Dai-Freed anomalies in particle physics, JHEP 08 (2019) 003, [arXiv:1808.00009].
  * (23) J. Davighi, B. Gripaios, and N. Lohitsiri, Global anomalies in the Standard Model(s) and Beyond, JHEP 07 (2020) 232, [arXiv:1910.11277].
  * (24) C.-T. Hsieh, Discrete gauge anomalies revisited, arXiv:1808.02881.
  * (25) J. Davighi and N. Lohitsiri, The algebra of anomaly interplay, SciPost Phys. 10 (2021), no. 3 074, [arXiv:2011.10102].
  * (26) L. E. Ibanez and G. G. Ross, Discrete gauge symmetry anomalies, Phys. Lett. B 260 (1991) 291–295.
  * (27) C. Csaki and H. Murayama, Discrete anomaly matching, Nucl. Phys. B 515 (1998) 114–162, [hep-th/9710105].