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

# Noninvertible symmetries and anomalies from gauging $1$-form electric
centers

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

W

e devise a general method for obtaining $0$-form noninvertible discrete chiral
symmetries in $4$-dimensional $SU(N)/\mathbb{Z}_{p}$ and $SU(N)\times
U(1)/\mathbb{Z}_{p}$ gauge theories with matter in arbitrary representations,
where $\mathbb{Z}_{p}$ is a subgroup of the electric $1$-form center symmetry.
Our approach involves placing the theory on a three-torus and utilizing the
Hamiltonian formalism to construct noninvertible operators by introducing
twists compatible with the gauging of $\mathbb{Z}_{p}$. These theories exhibit
electric $1$-form and magnetic $1$-form global symmetries, and their
generators play a crucial role in constructing the corresponding Hilbert
space. The noninvertible operators are demonstrated to project onto specific
Hilbert space sectors characterized by particular magnetic fluxes.
Furthermore, when subjected to twists by the electric $1$-form global
symmetry, these surviving sectors reveal an anomaly between the noninvertible
and the $1$-form symmetries. We argue that an anomaly implies that certain
sectors, characterized by the eigenvalues of the electric symmetry generators,
exhibit multi-fold degeneracies. When we couple these theories to axions,
infrared axionic noninvertible operators inherit the ultraviolet structure of
the theory, including the projective nature of the operators and their
anomalies. We discuss various examples of vector and chiral gauge theories
that showcase the versatility of our approach.

##  1 Introduction

Symmetry is the bedrock upon which quantum field theory (QFT) is constructed.
In the past decade, a seismic shift has occurred in our understanding of
symmetries, transcending their conventional application to mere point-like
particles. In the contemporary paradigm, a $p$-form symmetry in $4$ dimensions
is linked to operators residing on $(3-p)$-dimensional topological manifolds
that act on $p$-dimensional objects charged under the symmetry. Moreover, over
the last couple of years, symmetries have expanded their domain to encompass
operators that defy the conventional notion of inversion. These are known as
noninvertible symmetries. While noninvertible symmetries initially found their
roots and applications in the realm of $2$-dimensional QFT, see, e.g.,
Fuchs:2002cm ; Komargodski:2020mxz , their significance in the context of
4-dimensional QFT sparked a deluge of research endeavors in this area (a non-
comprehensive list is Nguyen:2021yld ; Nguyen:2021naa ; Choi:2021kmx ;
Wang:2021vki ; Bhardwaj:2022yxj ; Choi:2022zal ; Kaidi:2022uux ; Choi:2022jqy
; Cordova:2022ieu ; Choi:2022rfe ; Bartsch:2022mpm ; Heckman:2022muc ;
Cordova:2022fhg ; Karasik:2022kkq ; GarciaEtxebarria:2022jky ; Choi:2022fgx ;
Yokokura:2022alv ; Bhardwaj:2022kot ; Bhardwaj:2022maz ; Bartsch:2022ytj ;
Heckman:2022xgu ; Apte:2022xtu ; Delcamp:2023kew ; Kaidi:2023maf ;
Putrov:2023jqi ; Dierigl:2023jdp ; Choi:2023pdp . Also, see Schafer-
Nameki:2023jdn ; Shao:2023gho  for reviews.)

It is well known that quantum electrodynamics has a classical $U(1)_{\chi}$
axial symmetry that breaks down because of the Adler–Bell–Jackiw (ABJ)
anomaly. However, it was realized in Choi:2022jqy ; Cordova:2022ieu  that the
axial symmetry does not completely disappear. Instead, it resurfaces as a
noninvertible symmetry for each fractional element of the classical
$U(1)_{\chi}$. This profound reinterpretation of symmetries prompted a
compelling quest to unearth analogous structures in QFT. In Anber:2023pny ,
one of the authors established a technique for unveiling noninvertible
$0$-form symmetries within $SU(N)\times U(1)$ gauge theories in the presence
of matter in representation ${\cal R}$. This approach employed the Hamiltonian
formalism, where the theory was put on a three-dimensional torus
$\mathbb{T}^{3}$, subjecting it to $\mathbb{Z}_{N}$ magnetic twists along all
three spatial directions. Taking the matter to be a single Dirac fermion, this
theory is endowed with invertible $\mathbb{Z}_{\scriptsize 2\mbox{gcd}(T_{\cal
R},d_{\cal R})}^{\chi}$ $0$-form chiral symmetry, where $T_{\cal R}$ and
$d_{\cal R}$ are the Dynkin index and dimension of ${\cal R}$, respectively.
Yet, it was shown that the theory also possesses a noninvertible
$\tilde{\mathbb{Z}}_{2T_{\cal R}}^{\chi}$ $0$-form chiral symmetry111In this
paper, a tilde is used to indicate that a given symmetry or operator is
noninvertible.. Such symmetry acts on the Hilbert space projectively by
selecting special sectors characterized by certain magnetic numbers. New
noninvertible symmetries were also revealed in Argurio:2023lwl  in theories
with mixed anomalies between $\mathbb{Z}_{2}^{(1)}$ $1$-form and $0$-form
discrete chiral symmetries.

The topological essence of symmetries, encompassing the noninvertible
variants, underscores their sensitivity to the global structure of the gauge
group. Consequently, the inquiry arises: how do we identify these
noninvertible symmetries within a general gauge group, characterized as either
$SU(N)/\mathbb{Z}_{p}$ or $SU(N)\times U(1)/\mathbb{Z}_{p}$ where
$\mathbb{Z}_{p}$ is a subgroup of the center symmetry? In this work, we answer
this question by devising a general method that applies to any theory with a
direct multiplication of abelian and semi-simple nonabelian gauge groups
quotiented by a discrete center, whether the theory is vector-like or chiral.
This is achieved by putting the theory on $\mathbb{T}^{3}$ and turning on
magnetic fluxes in a refined subgroup of $\mathbb{Z}_{N}$, depending on the
matter content as well as the global structure of the gauge group.

In the context of $SU(N)$ gauge theory, the introduction of matter
characterized by an $N$-ality $n$ has the effect of breaking the
$\mathbb{Z}_{N}$ center of the group down to a subgroup $\mathbb{Z}_{q}$,
where $q$ is the greatest common divisor (gcd) of $N$ and $n$. Our focus is on
understanding the noninvertible $0$-form symmetries present in the
$SU(N)/\mathbb{Z}_{p}$ gauge theories, where $\mathbb{Z}_{p}$ is a subgroup of
the remaining center $\mathbb{Z}_{q}$. These theories exhibit both electric
$\mathbb{Z}_{q/p}^{(1)}$ and magnetic $\mathbb{Z}_{p}^{(1)}$ 1-form global
symmetries222There are $p$ distinct theories
$\left(SU(N)/\mathbb{Z}_{p}\right)_{n}$, where $n=0,1,..,p-1$ are the discrete
$\theta$-like parameters Aharony:2013hda . These theories differ by the set of
compatible line operators (Wilson, ’t Hooft, and dyonic operator). Here, we
restrict our analysis to $n=0$.. To identify the noninvertible symmetries, we
initiate the process starting from $SU(N)$ theory endowed with a single Dirac
fermion in representation ${\cal R}$, which possesses an invertible
$\mathbb{Z}_{2T_{\cal R}}^{\chi}$ chiral symmetry. We then subject this theory
to electric and magnetic twists characterized by elements of $\mathbb{Z}_{p}$.
If the theory exhibits a mixed anomaly between its chiral and electric
$\mathbb{Z}_{p}^{(1)}$ 1-form symmetries, the act of gauging $\mathbb{Z}_{p}$
effectively reveals the chiral symmetry as noninvertible. The construction of
a gauge-invariant operator corresponding to the noninvertible symmetry
$\tilde{\mathbb{Z}}_{2T_{\cal R}}^{\chi}$ involves several steps. First, we
create a topological operator by integrating the anomalous current
conservation law over $\mathbb{T}^{3}$. The resulting operator is not
invariant under $\mathbb{Z}_{p}$ gauge transformations. Yet, we can restore
gauge invariance by summing over all possible $\mathbb{Z}_{p}$ gauge-
transformed operators. This process results in a noninvertible chiral symmetry
operator that projects onto specific sectors in the Hilbert space, each
characterized by certain ’t Hooft lines charged under the magnetic
$\mathbb{Z}_{p}^{(1)}$ 1-form symmetry. $\tilde{\mathbb{Z}}_{2T_{\cal
R}}^{\chi}$ can exhibit further anomalies when subjected to twists by the
electric $\mathbb{Z}_{q/p}^{(1)}$ $1$-form symmetry, implying that states
within the Hilbert space of the $SU(N)/\mathbb{Z}_{p}$ gauge theory will
display multiple degeneracies.

We employ a similar approach to identify noninvertible symmetries in
$SU(N)\times U(1)/\mathbb{Z}_{p}$ gauge theories, where $\mathbb{Z}_{p}$ is a
subgroup of the electric $\mathbb{Z}_{N}^{(1)}$ 1-form center symmetry. Unlike
in $SU(N)$ theories, the introduction of matter does not reduce the
$\mathbb{Z}_{N}$ center. This is due to the presence of an abelian $U(1)$
sector, which ensures that all matter representations adhere to the cocycle
condition. In addition to the $1$-form electric center symmetry, this theory
is also endowed with a magnetic $U^{(1)}_{m}(1)$ 1-form symmetry. $SU(N)$
gauge theory with matter exhibits an anomaly between its chiral and $U(1)$
baryon-number symmetries. Gauging the latter transforms the theory into an
$SU(N)\times U(1)$ gauge theory and reveals the chiral symmetry
$\tilde{\mathbb{Z}}_{2T_{\cal R}}^{\chi}$ as noninvertible. Placing the theory
on $\mathbb{T}^{3}$ enables us to construct the corresponding noninvertible
chiral operator by summing over large $U(1)$ gauge transformations with
distinct winding numbers. Furthermore, since the theory exhibits a $1$-form
electric center symmetry, we can decorate the noninvertible operator with
$\mathbb{Z}_{N}$ magnetic twists. If we choose to further gauge a
$\mathbb{Z}_{p}^{(1)}$ subgroup of the electric $\mathbb{Z}_{N}^{(1)}$
symmetry, thereby resulting in the $SU(N)\times U(1)/\mathbb{Z}_{p}$ theory,
we must ensure that the noninvertible operator remains invariant under
$\mathbb{Z}_{p}$ gauge transformation. This is accomplished by summing over
all $\mathbb{Z}_{p}$ gauge-transformed chiral operators. Once again, we
discover that the resultant operator projects onto specific sectors within the
Hilbert space, distinguished by the presence of ’t Hooft lines charged under
$U^{(1)}_{m}(1)$. The noninvertible symmetry also exhibits a mixed anomaly
with the remaining electric $\mathbb{Z}_{N/p}^{(1)}$ global symmetry. The
anomaly implies that certain sectors of the theory, designated by certain
$\mathbb{Z}_{N/p}^{(1)}$ electric fluxes, exhibit multi-fold degeneracy.

Placing the theory on $\mathbb{T}^{3}$ offers a distinct advantage: it
presents a systematic approach for computing the ’t Hooft anomalies inherent
to a given theory. Simultaneously, it provides a means to construct the
Hilbert space explicitly. In our work, we put a significant emphasis on this
Hilbert space construction, shedding light on the intricate relationship
between Wilson’s lines, ’t Hooft lines, and the noninvertible operator.
Specifically, through several illustrative examples, we showcase how the
noninvertible chiral operator, within the framework of the Hilbert space and
Hamiltonian formalism, acts to annihilate the minimal ’t Hooft lines.

We also introduce couplings of gauge theories to axions. The underlying
renormalization group invariance of the noninvertible symmetries, along with
their associated anomalies, guarantees that the infrared (IR) axion physics
faithfully inherits all the characteristics of the theory at the ultraviolet
(UV) level. We substantiate this by explicitly constructing noninvertible
chiral operators, commencing from the IR anomalous axion current conservation
law. In our exploration, we offer concrete illustrations of various UV
theories and their corresponding IR axion physics manifestations.

This paper is organized as follows. In Section 2, we provide a concise
overview of the essential elements required for the development of
noninvertible symmetries. This section encompasses the introduction of our
notation, a review of the path-integral formalism on the 4-torus
($\mathbb{T}^{4}$), ’t Hooft twists, and the Hamiltonian formalism on
$\mathbb{T}^{3}$. Moving on to Section 3, we proceed to construct
noninvertible symmetries within the context of $SU(N)/\mathbb{Z}_{p}$ theories
while also identifying their associated anomalies. This section concludes with
the presentation of specific examples of noninvertible symmetries in both
vector and chiral gauge theories. In Section 4, we replicate the same
analysis, this time focusing on $SU(N)\times U(1)/\mathbb{Z}_{p}$. Two
examples are discussed, including the Standard Model (SM), and we demonstrate
that the SM lacks noninvertible symmetries within its non-gravitational
sector. Finally, our paper culminates in Section 5, where we explore the
coupling of gauge theories to axions. We show that noninvertible symmetry
operators can also be constructed using the axion anomalous current.

##  2 Preliminaries

In this section, we review the path integral and the Hamiltonian formalisms of
gauge theories put on a compact manifold with possible ’t Hooft twists, both
in space and time directions. Additionally, we examine the global symmetries
and anomalies in both formalisms, providing an exploration of these key
aspects. We base our formalism and notation on tHooft:1981sps ; vanBaal:1984ra
; Cox:2021vsa ; Anber:2023pny , and set the stage for constructing the
noninvertible operators we carry out in the subsequent sections. While some
results in this section are new, many are a mere review of previous results.
Moreover, some details are avoided, referring the reader to the literature for
an in-depth discussion. Yet, the information encapsulated here is necessary to
make this paper self-contained.

###  2.1 Twisting in the Path integral

#### Pure $SU(N)$ theory

We begin by reviewing ’t Hooft twists on a compact $4$-dimensional Euclidean
manifold with nontrivial $2$-cycles. We consider $SU(N)$ pure Yang-Mills (YM)
theory on $\mathbb{T}^{4}$, where $\mathbb{T}^{4}$ is a $4$-torus with periods
of length $L_{\mu}$, $\mu=1,2,3,4$333YM theory on $\mathbb{T}^{4}$ with
$\mathbb{Z}_{N}$ ’t Hooft twists dates back to the original work by ’t Hooft
tHooft:1981sps . The groundbreaking paper Gaiotto:2017yup  unveiled a novel
mixed anomaly, specifically involving the electric $\mathbb{Z}_{N}^{(1)}$
$1$-form symmetry. The background of the $1$-form symmetry is a $2$-form field
that can be implemented via a ’t Hooft twist. This fact led to a wave of
enthusiasm to understand the semi-classical limit of gauge theories on
$\mathbb{T}^{4}$ or $\mathbb{T}^{2}\times\mathbb{R}^{2}$ Anber:2022qsz ;
Anber:2023sjn ; Poppitz:2022rxv ; Tanizaki:2022ngt .. The $SU(N)$ gauge fields
$A_{\mu}$ are taken to obey the boundary conditions

$\displaystyle A_{\nu}(x+L_{\mu}\hat{e}_{\mu})=\Omega_{\mu}\circ
A_{\nu}(x)\equiv\Omega_{\mu}(x)A_{\nu}(x)\Omega_{\mu}^{-1}(x)-i\Omega_{\mu}(x)\partial_{\nu}\Omega_{\mu}^{-1}(x)\,,$
(1)

upon traversing $\mathbb{T}^{4}$ in each direction. The transition functions
$\Omega_{\mu}$ are $N\times N$ unitary matrices in the defining representation
of $SU(N)$, and $\hat{e}_{\nu}$ are unit vectors in the $x_{\nu}$ direction.
The subscript $\mu$ in $\Omega_{\mu}$ means that the function $\Omega_{\mu}$
does not depend on the coordinate $x_{\mu}$. Then, the compatibility of (1) at
the corners of the $x_{\mu}-x_{\nu}$ plane of $\mathbb{T}^{4}$ gives the
cocycle condition

$\displaystyle\Omega_{\mu}(x+\hat{e}_{\nu}L_{\nu})\;\Omega_{\nu}(x)=e^{i{2\pi
n_{\mu\nu}\over N}}\Omega_{\nu}(x+\hat{e}_{\mu}L_{\mu})\;\Omega_{\mu}(x)\,.$
(2)

The exponent $e^{i{2\pi n_{\mu\nu}\over N}}$, with anti-symmetric integers
$n_{\mu\nu}=-n_{\nu\mu}$, is the $\mathbb{Z}_{N}$ center of $SU(N)$. The
freedom to twist by elements of the center stems from the fact that both the
transition function and its inverse appear in (1). This is also equivalent to
the fact that the Wilson lines in pure $SU(N)$ gauge theory are charged under
the electric $\mathbb{Z}_{N}^{(1)}$ $1$-form center symmetry. The fundamental
(defining representation) Wilson lines wind around the $4$ cycles and are
given by

$\displaystyle
W_{\mu}=\mbox{tr}_{\Box}\left[Pe^{i\int_{x_{\mu}=0}^{x_{\mu}=L_{\mu}}A_{\mu}}\Omega_{\mu}\right]\,,$
(3)

where $\Box$ denotes the defining representation of $SU(N)$ and the insertion
of the transition function $\Omega_{\mu}$ ensures the gauge invariance of the
lines. It will be useful to break $n_{\mu\nu}$ into spatial (magnetic) $m_{i}$
and temporal (electric) $k_{i}$ twists:

$\displaystyle k_{i}\equiv n_{i4}\,,\quad n_{ij}\equiv\epsilon_{ijk}m_{k}\,,$
(4)

and $i,j=1,2,3$ or $x,y,z$. We also use bold-face letters, e.g.,
$\bm{k}\equiv(k_{1},k_{2},k_{3})$, to denote $3$-dimensional vectors. When
applied to the gauge fields on $\mathbb{T}^{4}$, the twists induce a
background with fractional topological charge444The simplest way to find the
topological charge is by activating the electric and magnetic ’t Hooft fluxes
along the Cartan generators of $SU(N)$; see, e.g., Anber:2019nze . We set
$F_{12}=-\frac{2\pi m_{3}}{L_{1}L_{2}}\nu_{a}H_{a}$ and $F_{34}=\frac{2\pi
k_{3}}{L_{3}L_{4}}\nu_{a}H_{a}$ along the $1$-$2$ and $3$-$4$ planes (and
similar expressions in the rest of the planes), where $H_{a}$ are the Cartan
generators, $\nu_{a}$ are the weights of the defining representation,
$a=1,...,N-1$, with summation over repeated indices. Plugging into
$Q=\frac{1}{8\pi^{2}}\int_{\mathbb{T}^{4}}\mbox{tr}[F\wedge F]$, and using
$\mbox{tr}[H_{a}H_{b}]=\delta_{ab}$ and $\nu_{a}\nu_{a}=1-1/N$, we find
$Q=\frac{\bm{k}\cdot\bm{m}}{N}+\mathbb{Z}$. tHooft:1981sps :

$\displaystyle Q=\frac{1}{8\pi^{2}}\int_{\mathbb{T}^{4}}\mbox{tr}[F\wedge
F]=-\frac{1}{8N}\epsilon_{\mu\nu\alpha\beta}n_{\mu\nu}n_{\alpha\beta}+\mathbb{Z}=\frac{\bm{k}\cdot\bm{m}}{N}+\mathbb{Z}\,,$
(5)

where $F$ is the field strength of $A$. Notice that the twists
$(\bm{m},\bm{k})\in(\mathbb{Z}\,\mbox{Mod}\,N)^{6}$. Adding multiples of $N$
to $\bm{m}$ or $\bm{k}$ leaves the cocycle condition intact. However, this has
the effect of changing the topological charges by integers. Hence, from here
on, we shall take the twists $m_{i},k_{i}\in\mathbb{Z}$, not
$\,\mbox{Mod}\,N$. The partition function of the $SU(N)$ gauge theory with
given twists $(\bm{m},\bm{k})$ is

$\displaystyle{\cal Z}[\bm{m},\bm{k}]_{\scriptsize
SU(N)}=\sum_{\nu\in\mathbb{Z}}\int\left[DA_{\mu}\right]_{(\bm{m},\bm{k})}e^{-S_{YM}-i\left(\frac{\bm{k}\cdot\bm{m}}{N}+\nu\right)\theta}\,.$
(6)

Here, $S_{YM}$ is the YM action, and the subscript $(\bm{m},\bm{k})$ indicates
that the path integral is to be performed with a given set of twisted boundary
conditions. Summation over the integer-valued topological sectors,
$\nu\in\mathbb{Z}$, is necessary so that the theory satisfies locality
(cluster decomposition).

####  $SU(N)$ theory with matter

Next, we add matter fields in a representation ${\cal R}$ under $SU(N)$. The
matter representation has $N$-ality $n$. Then, the full $\mathbb{Z}_{N}$
center breaks down to $\mathbb{Z}_{q}$, $q=\mbox{gcd}(N,n)$, i.e., the Wilson
lines are charged under $\mathbb{Z}_{q}^{(1)}$ $1$-from center symmetry555For
example, $SU(2M)$ gauge theory with matter in the $2$-index (anti)symmetric
representation has a $\mathbb{Z}_{2}^{(1)}$ center symmetry that acts on
Wilson lines.. Putting the matter, which, from now on, will be assumed to be
fermions, on $\mathbb{T}^{4}$ modifies the cocycle conditions. Let $\psi$ be a
left-handed Weyl fermion transforming under ${\cal R}$ of $SU(N)$. Then, the
fermion obeys the boundary conditions

$\displaystyle\psi(x+\hat{e}_{\mu}L_{\mu})={\cal
R}(\Omega_{\mu}(x))\psi(x)\,.$ (7)

The matrix ${\cal R}(\Omega_{\mu}(x))$ is built from $\Omega_{\mu}$,
transforming in the defining representation of $SU(N)$, with suitable
symmetrization or anti-symmetrization over $n$ indices (the $N$-ality of the
representation) according to the specific representation ${\cal R}$. Thus,
schematically (ignoring symmetrization over indices)

$\displaystyle{\cal
R}(\Omega_{\mu})\sim\underbrace{\Omega_{\mu}...\Omega_{\mu}}_{n}\,.$ (8)

${\cal R}(\Omega_{\mu})$ must satisfy the cocycle condition

$\displaystyle{\cal R}(\Omega_{\mu}(x+\hat{e}_{\nu}L_{\nu}))\;{\cal
R}(\Omega_{\nu}(x))={\cal R}(\Omega_{\nu}(x+\hat{e}_{\mu}L_{\mu}))\;{\cal
R}(\Omega_{\mu}(x))\,,$ (9)

which, via Eq. (2), reveals that the allowed values of the twists $\bm{m}$ and
$\bm{k}$ are $\frac{N}{q},\frac{2N}{q},...$. Twisting by the center subgroup
$\mathbb{Z}_{q}$ induces a background field with fractional topological charge

$\displaystyle
Q=\frac{\bm{m}\cdot\bm{k}}{N}+\mathbb{Z}\,,\quad\bm{m},\bm{k}\in\frac{N}{q}\mathbb{Z}\,,$
(10)

and the partition function in the presence of matter reads

$\displaystyle{\cal Z}[\bm{m},\bm{k}]_{\scriptsize SU(N)+\mbox{matter}}$
$\displaystyle=$
$\displaystyle\sum_{\nu\in\mathbb{Z}}\int\\{\left[DA_{\mu}\right]\left[D\mbox{matter}\right]\\}_{(\bm{m},\bm{k})}e^{-S_{YM}-S_{\scriptsize\mbox{matter}}-i\left(\frac{\bm{k}\cdot\bm{m}}{N}+\nu\right)\theta}\,,$
(11) $\displaystyle m_{i},k_{i}\in\frac{N}{q}\mathbb{Z}\,,\quad i=1,2,3\,.$

In the presence of matter, the theory is endowed with classical nonabelian and
abelian flavor symmetries. The $U(1)$ baryon-number symmetry survives the
quantum corrections. In contrast, the chiral part of the abelian symmetry,
denoted by $U(1)_{\chi}$, will generally break down to a discrete symmetry
because of the Adler–Bell–Jackiw (ABJ) anomaly of $U(1)_{\chi}$ in the
background of color instantons (which have integer topological charges). To
fix ideas, we consider a single flavor of a Dirac fermion with classical
$U(1)$ baryon number and $U(1)_{\chi}$ chiral symmetries. We take the $U(1)$
baryon charge of the Dirac fermion to be $+1$. The ABJ anomaly breaks
$U(1)_{\chi}$ down to invertible $\mathbb{Z}_{2T_{\cal R}}^{\chi}$ chiral
symmetry, where $T_{\cal R}$ is the Dynkin index of the representation.
Generalizing the theory to include many flavors is straightforward, and we
shall work out examples of this sort later in the paper. In the presence of
the twists $(\bm{m},\bm{k})$, there can be an anomaly of $\mathbb{Z}_{2T_{\cal
R}}^{\chi}$ in the background of $\mathbb{Z}_{q}^{(1)}$. The anomaly is a non-
trivial phase acquired by ${\cal Z}[\bm{m},\bm{k}]_{\scriptsize
SU(N)+\mbox{matter}}$ as we apply a transformation by an element of
$\mathbb{Z}_{2T_{\cal R}}$:

$\displaystyle{\cal Z}[\bm{m},\bm{k}]_{\scriptsize
SU(N)+\mbox{matter}}|_{k_{i},m_{i}\in N\mathbb{Z}/q}\longrightarrow
e^{i2\pi\ell\frac{\bm{m}\cdot\bm{k}}{N}}{\cal Z}[\bm{m},\bm{k}]_{\scriptsize
SU(N)+\mbox{matter}}\,,$ (12)

and $\ell=0,1,2,..,T_{\cal R}-1$ are the elements of $\mathbb{Z}_{2T_{\cal
R}}^{\chi}$. For the smallest twists $m_{j}=k_{j}=\frac{N}{q}$ in the $j$-th
direction, we obtain Anber:2021lzb

$\displaystyle{\cal Z}[\bm{m},\bm{k}]_{\scriptsize
SU(N)+\mbox{matter}}|_{m_{3}=k_{3}=\frac{N}{q}}\longrightarrow
e^{i2\pi\ell\frac{N}{q^{2}}}{\cal Z}[\bm{m},\bm{k}]_{\scriptsize
SU(N)+\mbox{matter}}\,.$ (13)

Bearing in mind that $N/q\in\mathbb{Z}$, we can generally absorb the integral
part of $N/q^{2}$ by adding an integer topological charge, which cannot change
the anomaly. Nevertheless, we will retain the phase as indicated in Eq. (13).
The phase is nontrivial, and hence there is an anomaly, if and only if
$\ell\frac{N}{q^{2}}\not\in\mathbb{Z}$. In the next section, we show how to
obtain the same anomaly using the Hamiltonian formalism.

We can do more regarding turning on fractional fluxes in $SU(N)$ with matter.
Instead of limiting ourselves to $\mathbb{Z}_{q}$ twists, we can twist with
the full $\mathbb{Z}_{N}$ center symmetry or any subgroup of it provided we
also turn on backgrounds of $U(1)$ baryon number symmetry Anber:2019nze ;
Anber:2021lzb . Let $\omega_{\mu}$ denote the $U(1)$ transition functions such
that for the $U(1)$ gauge field $a_{\mu}$, we have
$a_{\nu}(x+\hat{e}_{\mu})=\omega_{\mu}\circ a_{\nu}(x)\equiv
a_{\nu}(x)-i\omega_{\mu}^{-1}\partial_{\nu}\omega_{\mu}$. Then, $\Omega_{\mu}$
and $\omega_{\mu}$ obey the cocycle conditions:

$\displaystyle\Omega_{\mu}(x+\hat{e}_{\nu}L_{\nu})\;\Omega_{\nu}(x)=e^{i{2\pi
n_{\mu\nu}\over N}}\Omega_{\nu}(x+\hat{e}_{\mu}L_{\mu})\;\Omega_{\mu}(x)\,,$
$\displaystyle\omega_{\mu}(x+\hat{e}_{\nu}L_{\nu})\;\omega_{\nu}(x)=e^{-i{2\pi
nn_{\mu\nu}\over N}}\omega_{\nu}(x+\hat{e}_{\mu}L_{\mu})\;\omega_{\mu}(x)\,,$
(14)

where the $N$-ality of the matter representation is incorporated in the
abelian transition functions. The topological charges of both the nonabelian
center and abelian backgrounds read666Here, we use Footnote 4 along with
abelian field strengths
$f_{12}=\frac{2\pi}{L_{1}L_{2}}(\frac{n}{N}m_{3}+A_{3})$ and
$f_{34}=\frac{2\pi}{L_{3}L_{4}}(\frac{n}{N}k_{3}+B_{3})$ in the $1$-$2$ and
$3$-$4$ planes, and similar expressions in the rest of the planes.
Substituting into $Q_{u}=\int_{\mathbb{T}^{4}}\frac{f\wedge f}{8\pi^{2}}$, we
obtain the fractional $U(1)$ topological charge.

$\displaystyle Q_{SU(N)}=\frac{\bm{m}\cdot\bm{k}}{N}+\mathbb{Z}\,,\quad
Q_{u}=\left(\frac{n}{N}\bm{m}+\bm{A}\right)\cdot\left(\frac{n}{N}\bm{k}+\bm{B}\right)\,,\quad\bm{m},\bm{k},\bm{A},\bm{B}\in\mathbb{Z}^{3}\,.$
(15)

Here, $\bm{A},\bm{B}$ are arbitrary integral magnetic and electric quantum
numbers that we can always turn on since they leave the cocycle condition
intact.

####  $SU(N)\times U(1)$ theory with matter

We may also choose to make $U(1)$ dynamical, which entails summing over small
and large gauge transformations of $U(1)$, with the latter implementing
integer winding. This results in $SU(N)\times U(1)$ gauge theory with a Dirac
fermion in representation ${\cal R}$, with $N$-ality $N$ and baryon-charge
$+1$. In this case, the $U(1)$ instantons reduce $\mathbb{Z}_{2T_{\cal
R}}^{\chi}$ down to the genuine (invertible) symmetry
$\mathbb{Z}^{\chi}_{\scriptsize 2\mbox{gcd}(T_{\cal R},d_{\cal R})}$, and
$d_{\cal R}$ is the dimension of $\cal R$. The easiest way to see that is by
recalling the partition function under a $U(1)_{\chi}$ transformation acquires
a phase:

$\displaystyle\exp\left[i2\alpha T_{\cal
R}\int_{\mathbb{T}^{4}}\frac{\mbox{tr}\left(F\wedge
F\right)}{8\pi^{2}}+i2\alpha d_{\cal R}\int_{\mathbb{T}^{4}}\frac{f\wedge
f}{8\pi^{2}}\right]\,,$ (16)

where $f$ is the field strength of the $U(1)$ field. Recalling that for the
dynamical $SU(N)$ and $U(1)$ fields we have
$\int_{\mathbb{T}^{4}}\frac{\mbox{tr}\left(F\wedge
F\right)}{8\pi^{2}}\in\mathbb{Z},\int_{\mathbb{T}^{4}}\frac{f\wedge
f}{8\pi^{2}}\in\mathbb{Z}$, we conclude that only
$\mathbb{Z}^{\chi}_{\scriptsize 2\mbox{gcd}(T_{\cal R},d_{\cal R})}$ survises
the chiral transformation. The theory admits Wilson’s lines:

$\displaystyle
W_{\mu\,,SU(N)}=\mbox{tr}_{\Box}\left[Pe^{i\int_{x_{\mu}=0}^{x_{\mu}=L_{\mu}}A_{\mu}}\Omega_{\mu}\right]\,,\quad
W_{\mu\,,U(1)}=e^{-i\int_{x_{\mu}=0}^{x_{\mu}=L_{\mu}}a_{\mu}}\omega_{\mu}\,,$
(17)

which are charged under an electric $\mathbb{Z}_{N}^{(1)}$ $1$-form center
symmetry. In addition, the theory is endowed with a magnetic $U^{(1)}_{m}(1)$
$1$-form symmetry because of the absence of magnetic monopoles. For the sake
of completeness, we also give the partition function of $SU(N)\times U(1)$
theory with matter in the background of given $(\bm{m},\bm{k})$ fluxes:

$\displaystyle{\cal Z}[\bm{m},\bm{k}]_{\scriptsize SU(N)\times
U(1)+\mbox{matter}}$ $\displaystyle=$
$\displaystyle\sum_{\nu,\nu_{U(1)}\in\mathbb{Z}}\int\\{\left[DA_{\mu}\right]\left[Da_{\mu}\right]\left[D\,\mbox{matter}\right]\\}_{(\bm{m},\bm{k})}e^{-S_{YM}-S_{U(1)}-S_{\scriptsize\mbox{matter}}}\,,$
(18) $\displaystyle m_{i},k_{i}\in\mathbb{Z}\,\,,\quad i=1,2,3\,,$

and in addition to the $SU(N)$ integer topological charges $\nu$, we included
a sum over integer topological charges $\nu_{U(1)}$ of the $U(1)$ sector.

###  2.2 Twisting in the Hamiltonian formalism

#### Pure $SU(N)$ theory

Let us repeat the above discussion using the Hamiltonian formalism, starting
with pure $SU(N)$ YM theory (we use a hat to distinguish an operator in this
section.) To this end, we put the gauge theory on a spatial $3$-torus
$\mathbb{T}^{3}$ and apply the magnetic $\bm{m}$ twists along the $3$-spatial
directions. The transition functions in the defining representation along the
spatial directions, denoted by $\Gamma_{i}$, can be chosen to be constant
$N\times N$ matrices obeying the cocycle condition

$\displaystyle\Gamma_{i}\;\Gamma_{j}=e^{i{2\pi\epsilon_{ijk}m_{k}\over
N}}\Gamma_{j}\;\Gamma_{i}\,.$ (19)

Then, one can construct the states of the physical Hilbert space using the
temporal gauge condition $A_{0}=0$. The states can be written using the
“position” eigenstates of the gauge fields $A_{j}$, $j=1,2,3$ (or $i=x,y,z$)
as follows:

$\displaystyle|\psi\rangle_{\bm{m}}\equiv|A_{1},A_{2},A_{3}\rangle_{\bm{m}}\,,\quad\hat{A}_{j}|A_{1},A_{2},A_{3}\rangle_{\bm{m}}=A_{j}|A_{1},A_{2},A_{3}\rangle_{\bm{m}}\,,$
(20)

and the subscript $\bm{m}$ emphasizes that the Hilbert space is constructed in
the background of the magnetic twists. In writing Eq. (20), we have put many
details under the rug, and the reader is referred to tHooft:1981sps ;
vanBaal:1984ra ; Cox:2021vsa  for details. For example, notice that the gauge
fields $A_{i}$ need to respect the twisted boundary conditions (19), i.e.,
they transform according to (1) as we traverse any spatial direction on
$\mathbb{T}^{3}$. The theory admits $3$ fundamental Wilson lines wrapping the
three cycles of $\mathbb{T}^{3}$; these are given by (3) by restricting $\mu$
to the spatial directions. The Wilson lines are charged under the
$\mathbb{Z}_{N}^{(1)}$ $1$-form symmetry generated by three symmetry
generators $\hat{T}_{j}$, the Gukov-Witten operators, supported on co-
dimension $2$ surfaces. Thus, we have

$\displaystyle\hat{T}_{j}\hat{W}_{j}=e^{i\frac{2\pi}{N}}\hat{W}_{j}\hat{T}_{j}\,,$
(21)

and there are $\hat{W}_{j}^{e_{j}}$ distinct Wilson’s lines with $N$ distinct
$N$-alities $e_{j}=0,1,..,N-1$. The center-symmetry generators $\hat{T}_{i}$
are hard to construct explicitly. However, their explicit form is not
important to us. What is important is that they commute with the YM
Hamiltonian $\hat{H}$, and thus, $\hat{H}$ and $\hat{T}_{i}$ can be
simultaneously diagonalized. The physical states of the theory
$|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}$ are designated by the
eigenvalues of $\hat{T}_{i}$. It can be shown that the action of $\hat{T}_{i}$
on $|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}$ is given by

$\displaystyle\hat{T}_{j}|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}=e^{i\frac{2\pi}{N}e_{j}-i\theta\frac{m_{j}}{N}}|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}\,,$
(22)

where $e_{j},m_{j}\in\mathbb{Z}_{N}$ and the $\theta$ term ensures that
$\hat{T}_{i}^{N}|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}=e^{-i\theta
m_{j}}|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}$, and hence,
$\hat{T}_{i}^{N}$ works as a large gauge transformation. The combination
$e_{j}-\frac{\theta}{2\pi}m_{j}$ is the $\mathbb{Z}_{N}$ electric flux in the
$j$-th direction. This is justified as follows. Consider the state
$\hat{W}_{j}|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}$, obtained from
$|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}$ by the action of
$\hat{W}_{j}$. Using Eqs. (21, 22), we find
$\hat{T}_{j}\hat{W}_{j}|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}=e^{i\frac{2\pi}{N}(e_{j}+1)-i\theta\frac{m_{j}}{N}}\hat{W}_{j}|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}$.
Therefore, acting with $\hat{W}_{j}$ on the state
$|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}$ increases $e_{j}$ by one unit
in the $j$-th direction. Since $\hat{W}_{j}$ inserts an electric flux tube
winding in the $j$-th direction, the interpretation of $e_{j}$ as electric
flux follows. Notice also that because $\hat{T}_{j}$ and $\hat{H}$ can be
simultaneously diagonalized, we may label the states by the energy and the
electric flux:

$\displaystyle|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}\equiv|E,\bm{e}\rangle_{\bm{m}}\,,\quad\bm{e}\in\mathbb{Z}_{N}^{3}\,.$
(23)

It is worth spending some time to explain our notation in Eq. (23), as we
shall use this notation extensively in our paper. The physical state is
labeled by the eigenvalues of a set of commuting operators, here the energy
and the electric flux. The $SU(N)$ theory does not admit a $1$-form magnetic
symmetry, and thus, we cannot label the states by magnetic fluxes. Yet, we can
turn on a background magnetic flux $\bm{m}$, indicated as a subscript; all
physical quantities are calculated in this magnetic background. Also, we use
the letter $\bm{m}$ to denote the set of magnetic fluxes we can consistently
turn on. Here, we have $\bm{m}\in\mathbb{Z}^{3}$.

How can we make sense of the fractional topological charge (10) on
$\mathbb{T}^{3}$? We consider the product of $\mathbb{T}^{3}$ and the time
interval $[0,L_{4}]$ and consider the boundary conditions
$\hat{A}_{i}(t=L_{4})=C[\bm{k}]\circ\hat{A}_{i}(t=0)$, where $C[\bm{k}]$ is an
“improper gauge” transformation implementing a twist $\bm{k}\in\mathbb{Z}^{3}$
on the gauge fields by an element of the center777In fact, $C$ should be
designated by both $\bm{k}$ and the integral instanton number $\nu$; see
tHooft:1981sps . However, $\nu$ does not play a role in this work.. In the
presence of the magnetic twists $\bm{m}$, it can be shown that an application
of $C[\bm{k}]$ results in the topological charge (Pontryagin square)
tHooft:1981sps ; vanBaal:1984ra ; Cox:2021vsa :

$\displaystyle
Q[C[\bm{k}]]=\int_{\mathbb{T}^{3}}K(C\circ\hat{A})-K(\hat{A})=\frac{1}{24\pi^{2}}\int_{\mathbb{T}^{3}}\mbox{tr}\left[CdC^{-1}\right]^{3}=\frac{\bm{m}\cdot\bm{k}}{N}+\mathbb{Z}\,,$
(24)

where $K(\hat{A})$ is the topological current density operator
$K(\hat{A})=\frac{1}{8\pi^{2}}\mbox{tr}\left[\hat{A}\wedge\hat{F}-\frac{i}{3}\hat{A}\wedge\hat{A}\wedge\hat{A}\right]$,
or in terms of the components:
$\hat{K}^{\mu}(A)=\frac{1}{16\pi^{2}}\epsilon^{\mu\nu\lambda\sigma}\left(\hat{A}_{\nu}^{a}\partial_{\lambda}\hat{A}_{\sigma}^{a}-\frac{1}{3}f^{abc}\hat{A}_{\nu}^{a}\hat{A}_{\lambda}^{b}\hat{A}_{\sigma}^{c}\right)$.

####  $SU(N)$ theory with matter

Adding fermions of $N$-ality $n$ changes the center from $\mathbb{Z}_{N}$ to
$\mathbb{Z}_{q}$, $q=\mbox{gcd}(N,n)$, and the twists $(\bm{m},\bm{k})$ are
now in $\left(N\mathbb{Z}/q\right)^{6}$. Otherwise, all the steps used to put
the theory on $\mathbb{T}^{3}$ and construct the Hilbert space carry over. In
particular, $\hat{T}_{i}$ now are the generators of the $\mathbb{Z}_{q}^{(1)}$
$1$-form symmetry, and their action on the physical states in the Hilbert
space is given by888It is conceivable to introduce an additional label to
signify the distinct symmetries generated by different operators
$\hat{T}_{j}$. For instance, we could designate $\hat{T}_{N,j}$ as the
generator of $\mathbb{Z}_{N}^{(1)}$ and $\hat{T}_{q,j}$ as the generator of
$\mathbb{Z}_{q}^{(1)}$. Nonetheless, this approach may lead to increased
complexity in our expressions, and we opt not to pursue it. Instead, we will
explicitly specify the symmetry in question when discussing these distinct
operators. (now we turn off the $\theta$ angle as we can rotate it away via a
chiral transformation acting on the fermion)

$\displaystyle\hat{T}_{j}|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}=e^{i\frac{2\pi}{q}e_{j}}|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}\,,$
(25)

and the theory has $e_{j}=0,1,2,..,q-1$ electric flux sectors in each
direction $j=1,2,3$. The operators $\hat{T}_{j}$ act on the spatial Wilson
lines in the defining representation of $SU(N)$ as
$\hat{T}_{j}\hat{W}_{j}=e^{i\frac{2\pi}{q}}\hat{W}_{j}\hat{T}_{j}$, and there
are $q$ distinct Wilson’s lines $W_{j}^{e_{j}}$. The physical states
$|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}$ are simultaneous eigenstates
of the Hamiltonian and $\hat{T}_{j}$ since both operators commute. Thus, we
can write the physical states in the magnetic flux background
$\bm{m}\in\left(N\mathbb{Z}/q\right)^{3}$ as

$\displaystyle|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}=|E,\bm{e}N/q\rangle_{\bm{m}}\,,\quad\bm{e}=(e_{1},e_{2},e_{3})\in\mathbb{Z}_{q}^{3}\,,$
(26)

and $Ne_{j}/q$ is the amount of electric flux carried by the state in
direction $j$. We may also say that $e_{j}$ is the number of electric fluxes
in units of $N/q$. For matter with $N$-ality $n=0$, e.g., in the adjoint
representation, $q=N$ and we recover what we have said about pure $SU(N)$
gauge theory.

The partition function (11) can be written in the Hamiltonian formalism as a
trace over states in Hilbert space:

$\displaystyle{\cal Z}[\bm{m},\bm{k}]_{\scriptsize SU(N)+\mbox{matter}}$
$\displaystyle=$
$\displaystyle\mbox{tr}_{\bm{m}}\left[e^{-L_{4}\hat{H}}(\hat{T}_{x})^{k_{x}}(\hat{T}_{y})^{k_{y}}(\hat{T}_{z})^{k_{z}}\right]$
(27) $\displaystyle=$
$\displaystyle\sum_{\bm{e}\in\\{0,1,..,q-1\\}^{3}}e^{i\frac{2\pi\bm{e}\cdot\bm{k}}{q}}{}_{\bm{m}}\langle
E,\bm{e}N/q|e^{-L_{4}\hat{H}}|E,\bm{e}N/q\rangle_{\bm{m}}\,,$

where the subscript $\bm{m}$ in the trace means that we are considering the
states in the background of the magnetic flux
$\bm{m}\in\left(N\mathbb{Z}/q\right)^{3}$. We also used Eqs. (25, 26), the
fact that the states are eigenstates of both the energy and the $1$-form
center operators.

To detect the anomaly between $\mathbb{Z}_{2T_{\cal R}}^{\chi}$ and
$\mathbb{Z}_{q}^{(1)}$ in the Hamiltonian formalism, we first define the
operator that implements the discrete chiral symmetry. To this end, we recall
that under a chiral $U(1)_{A}$ rotation, the presence of the ABJ anomaly
indicates non-conservation of the corresponding symmetry rotation:

$\partial_{\mu}\hat{j}^{\mu}_{A}=2T_{\cal R}\partial_{\mu}\hat{K}^{\mu}(A)\,.$
(28)

Yet, we can define a conserved current:

$\displaystyle\hat{j}_{5}^{\mu}\equiv\hat{j}_{A}^{\mu}-2T_{\cal
R}\hat{K}^{\mu}\,,$ (29)

and correspondingly a conserved charge:

$\displaystyle\hat{Q}_{5}=\int_{\mathbb{T}^{3}}\hat{J}_{5}^{0}\,.$ (30)

Therefore, it is natural to define the operator

$\displaystyle\hat{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}\equiv\exp\left[i\frac{2\pi\ell}{2T_{\cal
R}}\hat{Q}_{5}\right]=\exp\left[i\frac{2\pi\ell}{2T_{\cal
R}}\int_{\mathbb{T}^{3}}(\hat{j}_{A}^{0}-2T_{\cal
R}\hat{K}^{0}(\hat{A}))\right]\,,$ (31)

for $\ell=0,1,..,T_{\cal R}-1$, which implements the action of the
$\mathbb{Z}_{2T_{\cal R}}^{\chi}$ chiral symmetry.
$\hat{U}_{\mathbb{Z}_{2T_{\cal R}}}$ is invariant under both small and large
$SU(N)$ gauge transformations (with integer winding). To find the mixed
anomaly between $\mathbb{Z}_{2T_{\cal R}}^{\chi}$ and $\mathbb{Z}_{q}^{(1)}$,
we compute the commutation between $\hat{T}_{j}$, which implements the action
of the electric center symmetry in the $j$-th direction, and
$\hat{U}_{\mathbb{Z}_{2T_{\cal R}}}$:

$\displaystyle\hat{T}_{j}\hat{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}\hat{T}_{j}^{-1}\,,$ (32)

remembering that the theory is in the background of a magnetic twist
$m_{j}\in\frac{N\mathbb{Z}}{q}$ in the $j$-th direction999Similar to the
Footnote 8, we could use a label that denotes the specific magnetic flux
background when we are dealing with the operator
$\hat{U}_{\mathbb{Z}_{2T_{\cal R}},\ell}$. This background can be taken in
sets such as $\frac{N\mathbb{Z}}{q}$ or $\frac{N\mathbb{Z}}{p}$, among others.
However, adopting this approach may introduce unnecessary complexity to our
notation. As a result, we have chosen to adopt a more transparent approach: we
will explicitly mention the magnetic flux background whenever we discuss this
operator.. First, $\hat{T}_{j}$ commutes with the current $j_{A}^{0}$ since
the latter is a color singlet operator. However, $\hat{K}^{0}$ fails to
commute with $\hat{T}_{j}$; the commutation between the two operators is found
by recalling that the action of $\hat{T}_{j}$ is implemented on the gauge
fields $\hat{A}_{j}$ as $\hat{A}_{j}=C[k_{j}]\circ\hat{A}_{j}$. Thus, we find,
after making use of (24),

$\displaystyle\hat{T}_{j}\exp\left[i2\pi\ell\int_{\mathbb{T}^{3}}\hat{K}^{0}(\hat{A})\right]\hat{T}_{j}^{-1}$
(33) $\displaystyle=$
$\displaystyle\exp\left[i2\pi\ell\int_{\mathbb{T}^{3}}\hat{K}^{0}\left(C[k_{j}]\circ\hat{A}\right)-\hat{K}^{0}(\hat{A})\right]\exp\left[i2\pi\ell\int_{\mathbb{T}^{3}}\hat{K}^{0}(\hat{A})\right]$
$\displaystyle=$
$\displaystyle\exp\left[i2\pi\ell\frac{m_{j}k_{j}}{N}\right]_{m_{j},k_{j}\in\left(\frac{N\mathbb{Z}}{q}\right)^{2}}\exp\left[i2\pi\ell\int_{\mathbb{T}^{3}}\hat{K}^{0}(\hat{A})\right]\,,$

noting the restriction $m_{j},k_{j}\in\left(\frac{N\mathbb{Z}}{q}\right)^{2}$
due to the presence of matter; otherwise, we would not satisfy the cocycle
condition. Collecting everything and using the minimal twists
$m_{j}=k_{j}=\frac{N}{q}$ we conclude

$\displaystyle\hat{T}_{j}\hat{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}\hat{T}_{j}^{-1}=e^{i2\pi\ell\frac{N}{q^{2}}}\hat{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}\,,$ (34)

which is exactly the mixed anomaly between the $\mathbb{Z}_{2T_{\cal
R}}^{\chi}$ chiral and the $\mathbb{Z}_{q}^{(1)}$ $1$-form center symmetries
found in (13) from the path integral formalism. The anomaly along with the
commutation relations (remember that both $\mathbb{Z}_{q}^{(1)}$ and
$\mathbb{Z}^{\chi}_{2T_{\cal R}}$ are good symmetries of the theory, and
hence, the corresponding operators commute with the Hamiltonian)

$\displaystyle[\hat{H},\hat{T}_{j}]=0\,,\quad[\hat{H},\hat{U}_{\mathbb{Z}_{2T_{\cal
R}}}]=0\,,$ (35)

furnishes a finite-dimensional space with a minimum dimension of
$q^{2}/\mbox{gcd}(q^{2},N)$. This means that sectors in Hilbert space exhibit
a $q^{2}/\mbox{gcd}(q^{2},N)$-fold degeneracy.

####  $SU(N)\times U(1)$ theory with matter

Next, we discuss the Hamiltonian quantization of $SU(N)\times U(1)$ gauge
theory with matter fields on $\mathbb{T}^{3}$ in the background of twists. In
this case, we may twist with the full $\mathbb{Z}_{N}$ center symmetry
provided we also turn on a background of $U(1)$. Thus, we replace the cocycle
conditions (19) with

$\displaystyle\Gamma_{i}\;\Gamma_{j}$ $\displaystyle=$ $\displaystyle
e^{i{2\pi\epsilon_{ijk}m_{k}\over N}}\Gamma_{j}\;\Gamma_{i}\,,$
$\displaystyle\omega_{i}(x+\hat{e}_{j}L_{j})\;\omega_{j}(x)$ $\displaystyle=$
$\displaystyle e^{-i{2\pi n\epsilon_{ijk}m_{k}\over
N}}\omega_{j}(x+\hat{e}_{i}L_{i})\;\omega_{i}(x)\,,$ (36)

and we included the $N$-ality of the matter representation $n$ in the cocycle
condition of the abelian field. This guarantees that the combined transition
functions satisfy the correct cocycle conditions in the presence of matter.
Here, we can allow background center fluxes with
$(\bm{m},\bm{k})\in\mathbb{Z}^{6}$ for all matter representations, thanks to
the $U(1)$ gauge group. We also introduce the operators $\hat{T}_{j}$ for
$SU(N)$ and $\hat{t}_{j}$ for $U(1)$, $j=1,2,3$. The combinations
$\hat{T}_{j}\hat{t}_{j}$ are the generators of the electric
$\mathbb{Z}_{N}^{(1)}$ $1$-form global symmetry and act on the spatial Wilson
lines in (17) as:
$\hat{T}_{j}W_{j,SU(N)}=e^{i\frac{2\pi}{N}}W_{j,SU(N)}\hat{T}_{j}$ and
$\hat{t}_{j}W_{j,U(1)}=e^{-i\frac{2\pi}{N}}W_{j,U(1))}\hat{t}_{j}$. The action
of $\hat{t}_{j}$ is implemented on the gauge fields, as usual, by improper
gauge transformations of $\hat{a}_{j}$ as
$\hat{a}_{j}=c[k_{j}]\circ\hat{a}_{j}$, and amounts to applying $nk_{j}$ (Mod
$N$) electric twists (notice the appearance of the $N$-ality). Unlike
$\hat{T}_{j}$, the explicit form of $\hat{t}_{j}$ is simple:

$\displaystyle\hat{t}_{j}\equiv
e^{i\lambda_{j}(x)}\,,\quad\lambda_{j}(x)=\frac{-2\pi
n}{N}\frac{x_{j}k_{j}}{L_{j}}\,.$ (37)

Since $\mathbb{Z}_{N}^{(1)}$ is a good global symmetry, we can choose the
states in Hilbert space to be eigenstates of the $\mathbb{Z}_{N}^{(1)}$
generators $\hat{T}_{j}\hat{t}_{j}$:

$\displaystyle\hat{T}_{j}\hat{t}_{j}|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}=e^{i\frac{2\pi
e_{j}}{N}}|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}\,,$ (38)

where $e_{j}=0,1,...,N-1$. Notice that the states are constructed in the
“fractional” background magnetic flux $\bm{m}\in\mathbb{Z}^{3}$ (remember that
in principle $m_{i}\in\mathbb{Z}$ Mod $N$, and thus, it implements the
fractional magnetic twist. However, we can always add multiples of $N$ to
$m_{i}$ without affecting the cocycle conditions, and hence, we drop the Mod
$N$ restriction.) In addition, the theory has a magnetic $U(1)^{(1)}_{m}$
$1$-form global symmetry, which can be used to characterize the physical
states by an “integer” value of the magnetic flux. Therefore, a state in the
physical Hilbert space can be labeled as

$\displaystyle|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}=|E,\bm{e},\bm{N}\rangle_{\bm{m}}\,,\quad\bm{e}\in\mathbb{Z}_{N}^{3}\,,$
(39)

and $\bm{N}=(N_{x},N_{y},N_{z})\in\mathbb{Z}^{3}$ (not Mod $N$) label the
integral magnetic fluxes of the $U(1)$ gauge group. The partition function
(18) can be written as a trace over states in Hilbert space in
$(\bm{m},\bm{k})$ backgrounds as follows:

$\displaystyle{\cal Z}[\bm{m},\bm{k}]_{\scriptsize SU(N)\times
U(1)+\mbox{matter}}$ $\displaystyle=$
$\displaystyle\mbox{tr}_{\bm{m}}\left[e^{-L_{4}\hat{H}}(\hat{T}_{x}\hat{t}_{x})^{k_{x}}(\hat{T}_{y}\hat{t}_{y})^{k_{y}}(\hat{T}_{z}\hat{t}_{z})^{k_{z}}\right]$
$\displaystyle=$
$\displaystyle\sum_{\bm{e}\in\\{0,1,..,N-1\\}^{3},\bm{N}\in\mathbb{Z}^{3}}e^{i\frac{2\pi\bm{e}\cdot\bm{k}}{N}}{}_{\bm{m}}\langle
E,\bm{e},\bm{N}|e^{-L_{4}\hat{H}}|E,\bm{e},\bm{N}\rangle_{\bm{m}}\,.$

We also build the operator that corresponds to the chiral transformation. This
construction was detailed in Anber:2023pny , and we do not repeat it here.
Instead, we only give a synopsis of the derivation, which is needed in this
work. The anomaly equation of the chiral current is

$\displaystyle\partial_{\mu}\hat{j}^{\mu}_{A}-2T_{\cal
R}\partial_{\mu}\hat{K}^{\mu}(\hat{A})-\frac{2d_{\cal
R}}{8\pi^{2}}\epsilon_{\mu\nu\lambda\sigma}\partial^{\mu}\hat{a}^{\nu}\partial^{\lambda}\hat{a}^{\sigma}=0\,.$
(41)

Then, the chiral symmetry operator in the background of the $m_{j}$ magnetic
flux is given by

$\displaystyle\hat{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}=\exp\left[i\frac{2\pi\ell}{2T_{\cal R}}\hat{Q}_{5}\right]\,,$ (42)

where the conserved charge $\hat{Q}_{5}$ is given by

$\displaystyle\hat{Q}_{5}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}^{3}}d^{3}x\left[\hat{j}_{\chi}^{0}-2T_{\cal
R}K^{0}(\hat{A})-2\frac{d_{\cal
R}}{8\pi^{2}}\epsilon^{ijk}\hat{a}_{i}\partial_{j}\hat{a}_{k}\right]$ (43)
$\displaystyle+{d_{R}\over 4\pi}(N_{z}+{n\over
N}n_{z})\left[\int\limits_{0}^{L_{y}}{dy\over
L_{y}}\int\limits_{0}^{L_{z}}dz\hat{a}_{z}(x=0,y,z)+\int\limits_{0}^{L_{x}}{dx\over
L_{x}}\int\limits_{0}^{L_{z}}dz\hat{a}_{z}(x,y=0,z)\right]$
$\displaystyle+\sum\limits_{\scriptsize\mbox{cyclic}}(x\rightarrow
y\rightarrow z\rightarrow x)\,.$

The last term comes from carefully treating the boundary term implied from the
transition functions $\omega_{j}(x)$, since, unlike $\Gamma_{j}$, they depend
explicitly on $x_{j}$, see Anber:2023pny  for details. In addition to the
background flux $n_{j}$, which introduces the fractional winding number, we
also allow integer magnetic winding $N_{j}$. Under a transformation with
$\hat{t}_{j}$, the integral of the abelian Chern-Simons term
$\hat{K}^{0}(\hat{a})=\epsilon^{ijk}\hat{a}_{i}\partial_{j}\hat{a}_{k}$ in the
background of the integral $M_{j}$ and fractional $m_{j}$ magnetic fluxes
transforms as (recall (37))

$\displaystyle\hat{t}_{j}\exp\left[i\int_{\mathbb{T}^{3}}\hat{K}^{0}(\hat{a})\right]t_{j}^{-1}$
$\displaystyle=$
$\displaystyle\exp\left[i\int_{\mathbb{T}^{3}}\hat{K}^{0}(c\circ\hat{a})-i\int_{\mathbb{T}^{3}}\hat{K}^{0}(\hat{a})\right]\exp\left[i\int_{\mathbb{T}^{3}}\hat{K}^{0}(\hat{a})\right]$
(44)
$\displaystyle=\left(N_{j}+\frac{nn_{j}}{N}\right)\left(\frac{nk_{j}}{N}\right)\exp\left[i\int_{\mathbb{T}^{3}}\hat{K}^{0}(\hat{a})\right]\,.$

The reader will notice that we switched from the letter $\bm{m}$, which we use
to signify the set of fractional fluxes we can activate, e.g., here we have
$\bm{m}\in\mathbb{Z}^{3}$, to the letter $\bm{n}$, which is the actual number
of fractional magnetic fluxes we turn on. We shall use the same labeling
throughout the paper.

In the next sections, we use these constructions to argue that
$SU(N)/\mathbb{Z}_{p}$, $\mathbb{Z}_{p}\subseteq\mathbb{Z}_{q}$ as well as
$SU(N)\times U(1)/\mathbb{Z}_{p}$, $\mathbb{Z}_{p}\subseteq\mathbb{Z}_{N}$
enjoy a noninvertible $0$-form chiral symmetry, with a possible mixed anomaly
with the $1$-form center symmetry.

##  3 $SU(N)/\mathbb{Z}_{p}$, $\mathbb{Z}_{p}\subseteq\mathbb{Z}_{q}$
theories, noninvertible symmetries, and their anomalies

In this section, we direct our attention to YM theories featuring matter
fields residing in a particular representation ${\cal R}$ and characterized by
an $N$-ality $n$. Building upon the discussion in the preceding section, it is
established that $SU(N)$ gauge theories, when coupled to matter, exhibit an
electric $\mathbb{Z}_{q}^{(1)}$ $1$-form center symmetry (recall
$q=\mbox{gcd}(N,n)$). A notable maneuver within this framework involves the
gauging of $\mathbb{Z}_{q}^{(1)}$ or a subgroup of it, leading to
$SU(N)/\mathbb{Z}_{p}$ theory, $\mathbb{Z}_{p}\subseteq\mathbb{Z}_{q}$, whose
partition function is obtained by summing over integer and fractional
topological charge sectors. Thus, gauge transformations with fractional
winding numbers are part of the gauge structure, and well-defined operators
should be invariant under such gauge transformations. Here, we would like to
emphasize that there are $p$ distinct theories:
$\left(SU(N)/\mathbb{Z}_{p}\right)_{n}$, $n=0,1,...,p$, which differ by the
admissible genuine (electric, magnetic, or dyonic) line operators. In this
paper, we limit our treatment to $\left(SU(N)/\mathbb{Z}_{p}\right)_{n=0}$,
and whenever we mention $SU(N)/\mathbb{Z}_{p}$, we particularly mean
$\left(SU(N)/\mathbb{Z}_{p}\right)_{0}$. What happens to the invertible
$\mathbb{Z}_{2T_{\cal R}}^{\chi}$ discrete chiral symmetry of this theory? As
we shall discuss, this symmetry can stay invertible or become noninvertible,
depending on whether it exhibits a mixed anomaly with $\mathbb{Z}_{p}^{(1)}$
symmetry in the original $SU(N)$ theory.

###  3.1 $SU(N)/\mathbb{Z}_{q}$

We start by discussing noninvertible $0$-form chiral symmetries in
$SU(N)/\mathbb{Z}_{q}$ theories, i.e., theories obtained by gauging the full
electric $\mathbb{Z}_{q}^{(1)}$ $1$-form center symmetry. Such theories do not
possess global electric $1$-form symmetry; hence, there are no genuine
Wilson’s lines. This can be understood as follows. We start with pure $SU(N)$
gauge theory, which has an electric $\mathbb{Z}_{N}^{(1)}$ $1$-form symmetry
and admits the full spectrum of Wilson lines, i.e., it admits Wilson’s lines
with all $N$-alities $n=0,1,2,..,N-1$. Gauging a $\mathbb{Z}_{q}$ subgroup of
$\mathbb{Z}_{N}$, we obtain $SU(N)/\mathbb{Z}_{q}$ gauge theory. Now, the
spectrum of allowed Wilson lines must be invariant under $\mathbb{Z}_{q}$,
forcing us to remove those lines with $N$-alities that are not multiples of
$q$. The remaining lines in pure $SU(N)/\mathbb{Z}_{q}$ theory are charged
under an electric $\mathbb{Z}_{N/q}^{(1)}$ $1$-form symmetry; these are
$W_{j}^{qe_{j}}$, with $e_{j}=0,1,..,N/q-1$ and $W_{j}$ is Wilson’s line in
the defining representation of $SU(N)$. Finally, introducing matter with
$N$-ality $q$ means that those remaining lines can end on the matter and must
also be removed from the spectrum. This deprives $SU(N)/\mathbb{Z}_{q}$ gauge
theory with matter from all genuine Wilson’s lines.

Despite that $SU(N)/\mathbb{Z}_{q}$ theory with matter does not possess an
electric $1$-form symmetry, it is endowed with a magnetic
$\mathbb{Z}_{q}^{m(1)}$ $1$-form global symmetry. This can be understood,
again, starting from the pure $SU(N)/\mathbb{Z}_{q}$ theory. As we discussed
above, the pure theory has an electric $\mathbb{Z}_{N/q}^{(1)}$ $1$-form
symmetry. The magnetic dual of $SU(N)/\mathbb{Z}_{q}$ is
$SU(N)/\mathbb{Z}_{N/q}$, which admits a magnetic $\mathbb{Z}_{q}^{m(1)}$
$1$-form symmetry. The pure $SU(N)/\mathbb{Z}_{q}$ theory has $q$ distinct
magnetic fluxes (’t Hooft lines) in its spectrum. Let ${\cal T}_{j}$ be the ’t
Hooft line winding around direction $j$ in the defining representation of
$SU(N)$, i.e., it has $N$-ality $1$. Then, the pure $SU(N)/\mathbb{Z}_{q}$
theory possesses the following set of ’t Hooft lines ${\cal
T}_{j}^{n_{j}N/q}$, $n_{j}=0,1,..,q-1$ for $j=1,2,3$, which are mutually local
with the set of Wilson’s lines $W_{j}^{qe_{j}}$,
$e_{j}=0,1,..,N/q-1$101010This can be easily seen since ${\cal
T}_{j}^{n_{j}N/q}$ and $W_{j}^{qe_{j}}$ satisfy the Dirac quantization
condition.. Introducing electric matter removes all Wilson’s lines (as stated
above) but does not alter the magnetic symmetry. Thus, we conclude that
$SU(N)/\mathbb{Z}_{q}$ theory with matter possesses a magnetic
$\mathbb{Z}_{q}^{m(1)}$ $1$-form global symmetry acting on a set of ’t Hooft
lines ${\cal T}_{j}^{n_{j}N/q}$, $n_{j}=0,1,..,q-1$ for $j=1,2,3$.

We can label the states in the physical Hilbert space of
$SU(N)/\mathbb{Z}_{q}$ theory with matter by both energy and magnetic fluxes
since the Hamiltonian commutes with the generators of the magnetic
$\mathbb{Z}_{q}^{m(1)}$ $1$-form symmetry111111Recall that the allowed
magnetic twists in the $SU(N)$ theory with matter are
$\bm{m}\in(N\mathbb{Z}/q)^{3}$.:

$\displaystyle|\psi\rangle_{\scriptsize\mbox{phy}}=|E,\bm{n}N/q\rangle\,,\quad\bm{n}=(n_{x},n_{y},n_{z})\in(\mathbb{Z}_{q})^{3}\,.$
(45)

The partition function of these theories involves summing over sectors with
fractional topological charges $N\mathbb{Z}/q^{2}$ (use Eq.(10) and set
$k_{i}=m_{i}=N/q$), which can be written in the path-integral formalism as (we
set the vacuum angle $\theta=0$)

$\displaystyle{\cal Z}_{\scriptsize SU(N)/\mathbb{Z}_{q}+\mbox{matter}}$
$\displaystyle=$
$\displaystyle\sum_{\nu\in\mathbb{Z},(\bm{m},\bm{k})\in\left(N\mathbb{Z}/q\right)^{6}}\int\\{\left[DA_{\mu}\right]D\left[\mbox{matter}\right]\\}_{(\bm{m},\bm{k})}e^{-S_{YM}-S_{\scriptsize\mbox{matter}}}\,,$

or in the Hamiltonian formalism as

$\displaystyle{\cal Z}_{\scriptsize
SU(N)/\mathbb{Z}_{q}+\mbox{matter}}=\mbox{tr}\left[e^{-L_{4}\hat{H}}\right]=\sum_{\scriptsize\mbox{physical
states}}{}_{\scriptsize\mbox{phy}}\langle\psi|e^{-L_{4}\hat{H}}|\psi\rangle_{\scriptsize\mbox{phy}}\,.$
(47)

Our main task is to build a gauge invariant operator that implements the
$\mathbb{Z}^{\chi}_{2T_{\cal R}}$ chiral transformation in
$SU(N)/\mathbb{Z}_{q}$ theory with matter. To this end, we use the Hamiltonian
formalism of Section 2.2, dropping the hats from all operators to reduce
clutter. We also use $x,y,z$ to label the three spatial directions. For
$\ell\in\mathbb{Z}_{2T_{\mathcal{R}}}^{\chi}$, the chiral symmetry operator is
given by:

$U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}=e^{2\pi
i\frac{\ell}{2T_{\mathcal{R}}}\int_{\mathbb{T}^{3}}\left(j^{0}_{A}-2T_{\mathcal{R}}K^{0}(A)\right)}\,.$
(48)

This operator is invariant under large gauge transformations with integer
winding numbers. We will now gauge the $\mathbb{Z}_{q}^{(1)}$ one-form
symmetry. In $SU(N)/\mathbb{Z}_{q}$ gauge theory with matter, we sum over
arbitrary $\mathbb{Z}_{q}$ twists with fractional topological charges
$N\mathbb{Z}/q^{2}$. We consider the operator
$U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ in the presence of magnetic fluxes
$\bm{m}\in\left(N\mathbb{Z}/q\right)^{3}$ (these are the magnetic fluxes that
label the physical states in Eq. (45).) Let $T_{x}$ be the generator of an
electric $\mathbb{Z}_{q}$ center twist along the $x$ direction (i.e., a
$\mathbb{Z}_{q}$ gauge transformation), and we take it to have the minimal
twist of $N/q$. It acts on $U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ via
(recall the discussion around Eq. (33))

$T_{x}U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}T_{x}^{-1}=e^{-2\pi i\ell
Q}U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\ =e^{-2\pi
i\ell\frac{n_{x}N}{q^{2}}}U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\,,\quad
n_{x}\in\mathbb{Z}\,.$ (49)

$n_{x}$ counts the magnetic fluxes inserted in the $y$-$z$ plane in units of
$N/q$. Identical relations to (49) hold in the $y$ and $z$ directions. As we
saw in the previous section, if $\ell\frac{N}{q^{2}}\not\in\mathbb{Z}$, there
is a mixed ’t Hooft anomaly between the electric $\mathbb{Z}_{q}^{(1)}$
$1$-form center and the discrete chiral symmetries of $SU(N)$ theory with
matter. Eq. (49) implies that the operator
$U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ is not gauge invariant under a
$\mathbb{Z}_{q}$ gauge transformation as we attempt to gauge
$\mathbb{Z}_{q}^{(1)}$. We can remedy this problem and reconstruct a gauge-
invariant operator, denoted by $\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}}}$, by
summing over all $\mathbb{Z}_{q}$ gauge transformations generated by $T_{x}$,
$T_{y}$ and $T_{z}$:

$\displaystyle\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$
$\displaystyle\equiv\sum_{p_{x},p_{y},p_{z}\in\mathbb{Z}}(T_{x})^{p_{x}}(T_{y})^{p_{y}}(T_{z})^{p_{z}}U_{\mathbb{Z}_{2T_{\mathcal{R}}}}(T_{x})^{-p_{x}}(T_{y})^{-p_{y}}(T_{z})^{-p_{z}}$
$\displaystyle=U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\sum_{p_{x},p_{y},p_{z}\in\mathbb{Z}}e^{-2\pi
i\frac{\ell N}{q^{2}}\left(p_{x}n_{x}+p_{y}n_{y}+p_{z}n_{z}\right)}\equiv
U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\sum_{\bm{p}\in\mathbb{Z}^{3}}e^{-2\pi
i\frac{\ell N}{q^{2}}\bm{p}\cdot\bm{n}}$
$\displaystyle=U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\sum_{l_{x}\in\mathbb{Z}}\delta\left(\frac{n_{x}\ell
N}{q^{2}}-l_{x}\right)\sum_{l_{y}\in\mathbb{Z}}\delta\left(\frac{n_{y}\ell
N}{q^{2}}-l_{y}\right)\sum_{l_{z}\in\mathbb{Z}}\delta\left(\frac{n_{z}\ell
N}{q^{2}}-l_{z}\right)\,.$ (50)

In the first line, we included a sum over arbitrary powers of
$T_{x},T_{y},T_{z}$ to enforce the gauge invariance. Then, we used Eq. (49) in
going from the first to the second line and the Poisson resummation formula in
going from the second to the third line. Even though
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ is gauge invariant, it has no
inverse; it is, in general, a noninvertible operator that implements the
action of $\tilde{\mathbb{Z}}_{2T_{\mathcal{R}}}^{\chi}$, and we use a tilde
to denote the noninvertible nature of symmetries and their operators. The
noninvertibility stems from the fact that
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}}}$ works as a projector: the
insertion of this operator in the path integral of $SU(N)/\mathbb{Z}_{q}$
theory with matter projects onto specific topological charge sectors of
$SU(N)/\mathbb{Z}_{q}$, depending on $\ell$. This can be seen from the second
line in (50), which is a sum over Fourier modes that projects in and out
sectors, depending on their topological charge, upon acting on them. One can
see the projective nature of $\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$
by inserting it into the partition function (47):

$\displaystyle\langle\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\rangle=\sum_{\scriptsize\mbox{physical
states}}{}_{\scriptsize\mbox{phy}}\langle\psi|e^{-L_{4}\hat{H}}\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}|\psi\rangle_{\scriptsize\mbox{phy}}\,,$
(51)

and then using the physical states defined in Eq. (45). We find that
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ annihilates sectors with
$\frac{n_{x}\ell N}{q^{2}}\notin\mathbb{Z}$, etc. We remind that
$\frac{n_{x}N}{q^{2}}$ is the topological charge (see Eq. (10)), which we can
write as

$\displaystyle\frac{n_{x}N}{q^{2}}=\underbrace{\frac{n_{x}N}{q}}_{m_{x}}\underbrace{\frac{N}{q}}_{k_{x}}\frac{1}{N}\,,$
(52)

and, as we mentioned earlier and emphasize now, $n_{x}$ is the number of
magnetic fluxes in units of $N/q$. The same applies to the magnetic sectors in
the $y$ and $z$ directions. We conclude that
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ selects sectors in Hilbert
space with certain magnetic fluxes.

We can make the following observations about
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$:

  1. 1.

If $\ell\in q\mathbb{Z}$, $\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ is
invertible since in this case $\frac{n_{x,y,z}\ell N}{q^{2}}\in\mathbb{Z}$ for
all values of $n_{x},n_{y},n_{z}\in\mathbb{Z}$. The invertible subgroup of
$\tilde{\mathbb{Z}}_{2T_{\mathcal{R}}}^{\chi}$ is
$\mathbb{Z}_{2T_{\mathcal{R}}/q}^{\chi}$.

  2. 2.

If $\gcd(\ell N/q,q)=1$, then we must have $n_{x},n_{y},n_{z}\in q\mathbb{Z}$.
In other words, $\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ projects onto
untwisted flux sectors. In particular, in the sector given by
$n_{x},n_{y},n_{z}\in q\mathbb{Z}$, the symmetry operator
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ act invertibly for all
elements of the chiral symmetry $\ell=1,2,..,T_{\cal R}$.

  3. 3.

If $\gcd(\ell N/q,q)=a\neq 1$ and $\ell<q$, then let $q=aq^{\prime}$, and we
must have $n_{x,y,z}\in q^{\prime}\mathbb{Z}$.
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ projects onto background
fluxes with topological charge $Q\in\mathbb{Z}/q^{\prime}$, i.e. sectors that
have $\mathbb{Z}_{q^{\prime}}$ twists.

  4. 4.

The noninvertibility of $\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ can
be seen by multiplying the operator by its inverse to find

$\displaystyle\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\times\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}^{-1}\sim\sum_{\bm{p}\in\mathbb{Z}^{3}}e^{-2\pi
i\frac{\ell N}{q^{2}}\bm{p}\cdot\bm{n}}\equiv{\cal C}\,.$ (53)

${\cal C}$ is known as the condensation operator, which can be thought of as a
sum over topological surface operators
$\exp[-i\oint_{\mathbb{T}^{2}\subset\mathbb{T}^{3}}B^{(2)}]=\exp[-i2\pi\mathbb{Z}/q]$
wrapping the three $2$-cycles of $\mathbb{T}^{3}$, and $B^{(2)}$ is the
$2$-form field of the $\mathbb{Z}^{(1)}_{q}$ $1$-form symmetry.

We use the fact that $SU(N)/\mathbb{Z}_{q}$ theory possesses a magnetic
$\mathbb{Z}_{q}^{m(1)}$ $1$-form global symmetry to make one more observation.
Let ${\cal T}_{j}$ be ’t Hooft line of $N$-ality $1$ in direction $j$. Then,
the minimal ’t Hooft line in $SU(N)/\mathbb{Z}_{q}$ theory is ${\cal
T}_{j}^{N/q}$, i.e., it has $N$-ality $N/q$. The minimal line acts on a
physical state by increasing its magnetic flux by one in units of
$N/q$121212Similar to the discussion we had after Eq. (22), we can also
consider the generators of the magnetic $1$-form symmetry and argue that
${\cal T}_{j}^{N/q}$ inserts a magnetic flux $N/q$, as measured by the action
of the magnetic $1$-form symmetry on the state ${\cal
T}_{j}^{N/q}|\psi\rangle_{\scriptsize\mbox{phy}}$.. Now, let us take a theory
with $\mbox{gcd}(N/q,q)=1$ so that
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell=1}$ acts projectively on
certain states. Then, $|E,(n_{x}=q,n_{y}=q,n_{z}=q)N/q\rangle$ is one of the
physical states that survive under the action of
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell=1}$. We have ${\cal
T}_{x}^{N/q}|E,(q,q,q)N/q\rangle=|E,(q+1,q,q)N/q\rangle$. Thus, we immediately
see from Eq. (50) that

$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell=1}{\cal
T}_{x}^{N/q}|E,(q,q,q)N/q\rangle=\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell=1}|E,(q+1,q,q)N/q\rangle\rangle=0\,.$
(54)

We write this result as

$\displaystyle\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell=1}{\cal
T}_{j}^{N/q}=0\,,\quad j=x,y,z\,.$ (55)

In other words, the operator
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell=1}$ annihilates the minimal ’t
Hooft lines in this theory. It also annihilates all ’t Hooft lines ${\cal
T}_{j}^{n_{j}N/q}$, $n_{j}\neq 0$ Mod $q$. This is an alternative way to see
the projective nature of this operator.

###  3.2 $SU(N)/\mathbb{Z}_{p}$

Next, we discuss $SU(N)/\mathbb{Z}_{p}$ theory with matter with $N$-ality $n$,
and
$\mathbb{Z}_{p}\subseteq\mathbb{Z}_{q}=\mathbb{Z}_{\scriptsize\mbox{gcd}(N,n)}$.
The partition function of this theory is given by the path integral in Eq.
(3.1), now restricting the sum over the electric and magnetic twists
$(\bm{m},\bm{k})\in(N\mathbb{Z}/p)^{6}$. The theory possess an electric
$\mathbb{Z}_{q/p}^{(1)}$ $1$-form global symmetry. As before, $T_{x}$ is taken
to be the generator of the electric $\mathbb{Z}_{q}^{(1)}$ symmetry. Then, the
electric $\mathbb{Z}_{q/p}^{(1)}$ $1$-form global symmetry is generated by
$T_{x}^{p}$ (as well as $T_{y}^{p}$ and $T_{z}^{p}$). The theory has $q/p$
distinct Wilson’s lines $W_{j}^{e_{j}p}$, with $e_{j}=0,1,2,..,q/p-1$. These
lines are invariant under $\mathbb{Z}_{p}$, as they should be since
$\mathbb{Z}_{p}$ is gauged. The minimal admissible Wilson’s line $W_{j}^{p}$
carries one electric flux in units of $pN/q$. In the limiting case $p=q$, the
line $W_{j}^{p=q}$ coincides with the matter content and must be removed from
the spectrum of line operators. Therefore, in this case, the theory does not
possess a $1$-form electric symmetry, as discussed in the previous section.

In addition, the theory has a magnetic $\mathbb{Z}_{p}^{m(1)}$ $1$-form
symmetry. If ${\cal T}_{j}$ is the ’t Hooft line with $N$-ality $1$, then the
minimal admissible ’t Hooft line in the theory is ${\cal T}_{j}^{N/p}$, which
carries one magnetic flux in units of $N/p$. There are $p$ distinct ’t Hooft
lines in the theory ${\cal T}_{j}^{n_{j}N/p}$, $n_{j}=0,1,..,p-1$, which are
mutually local with Wilson’s lines $W_{j}^{e_{j}p}$. The Hamiltonian, Wilson’s
lines generators, and the ’t Hooft lines generators of this theory can be
simultaneously diagonalized. Therefore, the energies and eigenvalues of the
set of Wilson and ’t Hooft operators can be used to label the physical states
of Hilbert space:

$\displaystyle|\psi\rangle_{\scriptsize\mbox{phy}}=|E,\bm{e}pN/q,\bm{n}N/p\rangle\,,\quad\bm{e}\in(\mathbb{Z}_{q/p})^{3}\,,\bm{n}\in(\mathbb{Z}_{p})^{3}\,.$
(56)

Next, we need to build a gauge invariant chiral symmetry operator. Our
starting point, as usual, is the operator

$U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}=e^{2\pi
i\frac{\ell}{2T_{\mathcal{R}}}\int_{\mathbb{T}^{3}}\left(j^{0}_{A}-2T_{\mathcal{R}}K^{0}(A)\right)}$
(57)

taken in the presence of the fractional magnetic fluxes
$\bm{m}\in\left(N\mathbb{Z}/p\right)^{3}$, which label the Hilbert space in
Eq. (56). The operator $T_{x}^{q/p}$ generates the electric
$\mathbb{Z}_{p}^{(1)}$ $1$-form symmetry, which is gauged. In other words,
$T_{x}^{q/p}$ implements the twists $\bm{k}\in\left(N\mathbb{Z}/p\right)^{3}$.
In analogy with $SU(N)/\mathbb{Z}_{q}$ theories, we need to build gauge
invariants of the chiral symmetry operator using the building block
$T_{x}^{q/p}U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}T_{x}^{-q/p}$. To compute
this block, we use the discussion around Eq. (33), taking the minimal twist
$N/p$ generated by $T_{x}^{q/p}$, to obtain

$T_{x}^{q/p}U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}T_{x}^{-q/p}=e^{-2\pi
i\ell\frac{n_{x}N}{p^{2}}}U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\,,\quad
n_{x}\in\mathbb{Z}\,,$ (58)

and $n_{x}$ counts the magnetic fluxes in units of $N/p$. If
$\ell\frac{N}{p^{2}}\not\in\mathbb{Z}$, there is a mixed anomaly between
$\mathbb{Z}_{2T_{R}}^{\chi}$ and the electric $\mathbb{Z}_{p}^{(1)}$
symmetries in $SU(N)$ theory with matter, and we expect the chiral symmetry
becomes noninvertible upon gauging $\mathbb{Z}_{p}^{(1)}$. The corresponding
gauge invariant operator of the $\tilde{\mathbb{Z}}_{2T_{\mathcal{R}}}^{\chi}$
symmetry is then given by the summations

$\displaystyle\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$
$\displaystyle=\sum_{p_{x},p_{y},p_{z}\in\mathbb{Z}}(T_{x})^{qp_{x}/p}(T_{y})^{qp_{y}/p}(T_{z})^{qp_{z}/p}U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}(T_{x})^{-qp_{x}/p}(T_{y})^{-qp_{y}/p}(T_{z})^{-qp_{z}/p}$
$\displaystyle=U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\sum_{p_{x},p_{y},p_{z}\in\mathbb{Z}}e^{-2\pi
i\frac{\ell N}{p^{2}}\left(p_{x}n_{x}+p_{y}n_{y}+p_{z}n_{z}\right)}$
$\displaystyle=U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\sum_{l_{x}\in\mathbb{Z}}\delta\left(\frac{n_{x}\ell
N}{p^{2}}-l_{x}\right)\sum_{l_{y}\in\mathbb{Z}}\delta\left(\frac{n_{y}\ell
N}{p^{2}}-l_{y}\right)\sum_{l_{z}\in\mathbb{Z}}\delta\left(\frac{n_{z}\ell
N}{p^{2}}-l_{z}\right)\,.$ (59)

This noninvertible operator generalizes (50) to any
$\mathbb{Z}_{p}\subseteq\mathbb{Z}_{q}$, and it projects onto sectors with
finer topological charges than the sectors admissible by (50). This means
there exist sectors where $\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ act
invertibly for all $\ell=1,2,..,T_{\cal R}$ if and only if

$\displaystyle l_{x}=\frac{n_{x}N}{p^{2}}\in\mathbb{Z}\,,$ (60)

with similar conditions in the $y$ and $z$ directions. As special cases, we
may first set $p=q$ to readily cover (50). Also, setting $p=1$, the operator
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ becomes invertible, as can be
easily seen from the second line in (59). Notice that
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ does not act on Wilson’s
lines in this theory, as the noninvertible operator is built from
$(T_{j})^{qp_{j}/p}$ and its inverse; thus, one can push a Wilson line through
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ without being
affected131313Although we do not give the explicit form of $T_{j}$, it can be
thought of as an exponential of an integral of the chromoelectric field over a
$2$-dimensional submanifold; see Reinhardt:2002mb . A Wilson line would
acquire a phase as we push it past $T_{j}^{q/p}$ (we use
$[A^{a}_{j}(\bm{x},t),E^{b}_{k}(\bm{y},t)]=i\delta_{jk}\delta(\bm{x}-\bm{y})\delta_{ab}$,
where $a,b$ are the color indices, along with the Baker-Campbell-Hausdorff
formula). It also acquires the negative of the same phase as it is pushed past
$T_{j}^{-q/p}$. Therefore, the phases cancel out, and hence, the result in Eq.
(61).. We can write this observation as

$\displaystyle\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}W_{j}^{e_{j}p}=W_{j}^{e_{j}p}\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\,,\quad
e_{j}=0,1,2,..,q/p-1\,,\quad j=x,y,z\,.$ (61)

This is very different from the action of
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ on ’t Hooft lines, as we
discussed before.

The procedure employed to construct the noninvertible operator
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ contains an additional layer
of underlying physics. It is essential to keep in mind that this operator is
constructed in $SU(N)/\mathbb{Z}_{p}$ theory, where its creation involved a
sum over magnetic $\bm{m}\in\left(N\mathbb{Z}/p\right)^{3}$ and electric
$\bm{k}\in\left(N\mathbb{Z}/p\right)^{3}$ twists. These twists do not
encompass the entire range of permissible twists that can be applied. Recall
that the theory encompasses a global $\mathbb{Z}_{q/p}^{(1)}$ symmetry, which
affords us the opportunity to introduce the electric twists
$\bm{k}\in\left(pN\mathbb{Z}/q\right)^{3}$. Moreover, we can turn on magnetic
twists $\bm{m}\in\left(pN\mathbb{Z}/q\right)^{3}$, compatible with the cocycle
condition141414Recall that twists in $N\mathbb{Z}/q$ are compatible with the
cocycle conditions. Therefore, twists in $pN\mathbb{Z}/q$ are a subset of the
allowed twists. Notice that the twists $\bm{m}\in(pN\mathbb{Z}/q)^{3}$ provide
background magnetic fluxes and do not label the physical states in Hilbert
space, Eq. (56).. This broader scope of twists provides a richer set of
possibilities within the theory. We recall that $T_{x}^{p}$ is the generator
of $\mathbb{Z}_{q/p}^{(1)}$ symmetry that implements the twists $k_{x}\in
pN\mathbb{Z}/q$. Then, one can write the partition function of
$SU(N)/\mathbb{Z}_{p}$ theory in these background twists as

$\displaystyle{\cal Z}_{\scriptsize
SU(N)/\mathbb{Z}_{p}+\mbox{matter}}[\bm{m},\bm{k}]$ $\displaystyle=$
$\displaystyle\mbox{tr}_{\bm{m}\in\left(pN\mathbb{Z}/q\right)^{3}}\left[e^{-L_{4}H}T_{x}^{k_{x}p}T_{y}^{k_{y}p}T_{z}^{k_{z}p}\right]$
(62) $\displaystyle=$
$\displaystyle\sum_{\bm{e}\in(\mathbb{Z}_{q/p})^{3}}e^{-i2\pi\frac{p\bm{k}\cdot\bm{e}}{q}}{}_{\scriptsize\mbox{phy}}\langle\psi|e^{-L_{4}H}|\rangle_{\scriptsize\mbox{phy}}|_{\bm{m}\in\left(pN\mathbb{Z}/q\right)^{3}}\,,$

and we used Eq. (56) along with
$T_{j}^{k_{j}p}|\psi\rangle_{\scriptsize\mbox{phy}}=e^{-i2\pi\frac{pk_{j}e_{j}}{q}}|\psi\rangle_{\scriptsize\mbox{phy}}$;
see the discussion around Eqs. (65, 66) below.

Next, consider the commutation relation between $T_{x}^{p}$ and
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$, the latter operator is being
in the background of the magnetic twist
$\bm{m}\in\left(pN\mathbb{Z}/q\right)^{3}$. Using the discussion and procedure
around Eq. (33), we obtain

$\displaystyle
T_{x}^{p}\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}T_{x}^{-p}=e^{-2\pi
i\ell
n_{x}\frac{p^{2}N}{q^{2}}}\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\,.$
(63)

The failure of the commutation between $T_{x}^{p}$ and
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ by the phase $e^{-2\pi i\ell
n_{x}\frac{p^{2}N}{q^{2}}}$, assuming $\ell
n_{x}\frac{p^{2}N}{q^{2}}\notin\mathbb{Z}$ , signals a mixed anomaly between
the noninvertible $\tilde{\mathbb{Z}}_{2T_{\mathcal{R}}}^{\chi}$ chiral
symmetry and the electric $\mathbb{Z}_{q/p}^{(1)}$ $1$-form global symmetry.
This anomaly means that certain sectors in Hilbert space exhibit degeneracy.
Let us analyze this situation more closely. We assume there exists a sector
with $n_{x},n{{}_{y}},n_{z}$ that satisfies Eq. (60), and thus, in this
sector, the symmetry operator $\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$
acts invertibly for all elements $\ell=1,2,..,T_{\cal R}$. Now,
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$, being a global symmetry
operator, commutes with the Hamiltonian:

$\displaystyle[\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell},H]=0\,.$ (64)

Likewise, since $\mathbb{Z}_{q/p}^{(1)}$ is a global symmetry, its generators
$T_{j}^{p}$ commute with the Hamiltonian:

$\displaystyle[T_{j}^{p},H]=0\,.$ (65)

This commutation relation, along with Eq. (25), implies that $T_{j}^{p}$ acts
on physical states in Hilbert space as (the label $\bm{l}=(l_{x},l_{y},l_{z})$
emphasizes that such states satisfy condition (60), such that
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ acts invertibly on such
states. Also, we suppressed the detailed dependence on the different quantum
numbers to reduce clutter)

$\displaystyle T_{j}^{p}|E,e_{j}\rangle_{\bm{l}}=e^{i\frac{2\pi
p}{q}e_{j}}|E,e_{j}\rangle_{\bm{l}}\,,$ (66)

and that the states are labeled by their energies as well as
$e_{j}=1,2,...,q/p$ distinct labels; these are the eigenvalues (fluxes) of the
$\mathbb{Z}_{q/p}^{(1)}$ symmetry operator. The algebra defined by the
commutation relations Eqs. (64, 65), along with the mixed anomaly represented
as Eq. (63), under the assumption of a nontrivial phase, furnishes a finite-
dimensional space with a minimum dimension of
$q^{2}/\mbox{gcd}(n_{x}p^{2}N,q^{2})$ (we take $n_{x}=n_{y}=n_{z}$). The
Hilbert space of physical states, which are labeled by $q/p$ distinct fluxes,
sit in $q^{2}/\mbox{gcd}(n_{x}p^{2}N,q^{2})$ orbits, and a rotation by
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell=1}$ links a state with a flux
$e_{j}$ to a state with a flux $e_{\scriptsize
j+\mbox{gcd}(n_{j}p^{2}N,q^{2})/(qp)}$ as:

$\displaystyle\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell=1}|E,e_{j}\rangle_{\bm{l}}=|E,e_{\scriptsize
j}+\mbox{gcd}(n_{j}p^{2}N,q^{2})/(qp)\rangle_{\bm{l}}\,.$ (67)

Using the commutation relation (64), one immediately sees that the states
$|E,e_{j}\rangle_{\bm{l}}$ and $|E,e_{\scriptsize
j}+\mbox{gcd}(n_{j}p^{2}N,q^{2})/(qp)\rangle_{\bm{l}}$ have the same
energy151515It is helpful to give a numerical example. Take $N=1000$, $q=500$,
and $p=20$. Such numbers are contrived and do not necessarily correspond to
any realistic theory. Condition (60) is satisfied if we take $n_{x}=2$. Then,
the phase in the anomaly Eq. (63) is $e^{-i2\pi/5}$, implying a $5$-fold
degeneracy. The theory has an electric $\mathbb{Z}^{(1)}_{25}$ $1$-form
symmetry, and thus, $25$ distinct flux states. These states set in $5$
different orbits such that the states labeled with $e_{1}$, $e_{6}$, $e_{11}$,
$e_{16}$, $e_{21}$ have the same energy, and the states
$e_{2}$,$e_{7}$,…,$e_{22}$, have the same energy, etc..

In the following subsections, we apply our formalism to examples of theories
with fermions in specific representations.

###  3.3 Examples

####  3.3.1 $SU(4n+2)/\mathbb{Z}_{2}$ and $SU(4n)/\mathbb{Z}_{2}$ with a Dirac
fermion in the $2$-index anti-symmetric representation

The $SU(4n+2)/\mathbb{Z}_{2}$ gauge theory with a $2$-index anti-symmetric
Dirac fermion ($N$-ality $2$) has a $\mathbb{Z}_{8n}^{\chi}$ chiral symmetry.
The $SU(4n+2)$ theory possesses an electric $\mathbb{Z}_{2}^{(1)}$ one-form
symmetry. In Argurio:2023lwl , the authors argued that upon gauging
$\mathbb{Z}_{2}^{(1)}$, the odd rotations of $\mathbb{Z}_{8n}^{\chi}$ become
non-invertible. We can show this is the case on $\mathbb{T}^{3}$ using our
construction. Setting $N=4n+2$ in (50), we obtain

$\tilde{U}_{\mathbb{Z}_{8n},\ell}=U_{\mathbb{Z}_{8n},\ell}\sum_{l_{x}\in\mathbb{Z}}\delta\left(\frac{n_{x}\ell}{2}-l_{x}\right)\sum_{l_{y}\in\mathbb{Z}}\delta\left(\frac{n_{y}\ell}{2}-l_{y}\right)\sum_{l_{z}\in\mathbb{Z}}\delta\left(\frac{n_{z}\ell}{2}-l_{z}\right)\,.$
(68)

For $\ell$ odd, $\tilde{U}_{\mathbb{Z}_{8n},\ell}$ projects onto untwisted
gauge sectors and becomes non-invertible.

The $SU(4n)/\mathbb{Z}_{2}$ theory with a $2$-index anti-symmetric Dirac
fermion has a $\mathbb{Z}_{8n-4}^{\chi}$ chiral symmetry. The cocycle
conditions, say in the $x$-direction, must satisfy (see Eq. (9))

$e^{2\pi i\frac{2n_{yz}}{4n}}=1\,.$ (69)

Therefore we must have $n_{yz}\in 2n\mathbb{Z}$. There is no mixed anomaly
between $\mathbb{Z}_{8n-4}^{\chi}$ and the electric $\mathbb{Z}_{2}^{(1)}$
symmetries in the $SU(4n)$ theory since the anomaly phase $\frac{n_{yz}}{2}\in
n\mathbb{Z}$ is trivial. Thus, the full chiral symmetry
$\mathbb{Z}_{8n-4}^{\chi}$ is invertible. This is also in agreement with
Argurio:2023lwl .

####  3.3.2 $SU(6)/\mathbb{Z}_{3}$ with a Dirac fermion in the $3$-index anti-
symmetric representation

This theory has a $\mathbb{Z}_{6}^{\chi}$ chiral symmetry. What is special
about this theory is that its bilinear fermion operator vanishes identically
because of Fermi statistics. Moreover, the $SU(6)$ theory exhibits a mixed
anomaly between its electric $\mathbb{Z}_{3}^{(1)}$ $1$-form center and chiral
symmetries Yamaguchi:2018xse ; Anber:2019nfu . Assuming confinement, then the
chiral symmetry must be broken in the infrared. Yet, this breaking has to be
accomplished via higher-order condensate. Using (50), we find that the
operator corresponding to a chiral transformation in $SU(6)/\mathbb{Z}_{3}$
theory is

$\tilde{U}_{\mathbb{Z}_{6},\ell}=U_{\mathbb{Z}_{6},\ell}\sum_{l_{x}\in\mathbb{Z}}\delta\left(\frac{2n_{x}\ell}{3}-l_{x}\right)\sum_{l_{y}\in\mathbb{Z}}\delta\left(\frac{2n_{y}\ell}{3}-l_{y}\right)\sum_{l_{z}\in\mathbb{Z}}\delta\left(\frac{2n_{z}\ell}{3}-l_{z}\right)\,.$
(70)

Hence, for $\ell\in\\{1,2,4,5\\}$, the operator
$\tilde{U}_{\mathbb{Z}_{6},\ell}$ projects onto untwisted gauge sectors, and
the chiral symmetry operator becomes noninvertible.

####  3.3.3 $2$-index $SU(6)$ chiral gauge theory

Our next example is a chiral gauge theory. This is $SU(6)$ YM theory with a
single left-handed Weyl fermion $\psi$ in the $2$-index symmetric
representation and $5$ flavors of left-handed Weyl fermions $\chi$ in the
complex conjugate $2$-index anti-symmetric representation. The fermion budget
ensures the theory is free from gauge anomalies. The theory encompasses
continuous global symmetry $SU(5)_{\chi}\times U(1)_{A}$, where $SU(5)_{\chi}$
acts on $\chi$. The charges of $\psi$ and $\chi$ under $U(1)_{A}$ are
$q_{\psi}=-5\,,q_{\chi}=2$. The theory is also endowed with a
$\mathbb{Z}_{4}^{\chi}$ 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 $SU(6)\times SU(5)_{\chi}$. It turns out, see Anber:2023yuh  for
details (also see Anber:2021iip ), that we must divide the global symmetry by
$\mathbb{Z}_{3}\times\mathbb{Z}_{5}$ to remove redundancies. Putting
everything together and remembering that the theory possesses an electric
$\mathbb{Z}_{2}^{(1)}$ $1$-form center symmetry (since all fermions have
$N$-ality $n=2$), we write the faithful global group as:

$\displaystyle G^{\scriptsize\mbox{g}}=\frac{SU(5)_{\chi}\times
U(1)_{A}}{\mathbb{Z}_{3}\times\mathbb{Z}_{5}}\times\mathbb{Z}_{4}^{\chi}\times\mathbb{Z}_{2}^{(1)}\,.$
(71)

This theory has an anomaly between its $\mathbb{Z}_{2}^{(1)}$ center symmetry
and $\mathbb{Z}_{4}^{\chi}$ chiral symmetry. To see the anomaly, we recall
that we can turn on the magnetic and electric twists
$(\bm{m},\bm{k})\in(3\mathbb{Z})^{6}$. This gives the topological charge
$Q\in\mathbb{Z}/2$. Thus, under a chiral transformation, the partition
function acquires a phase

$\displaystyle{\cal Z}[\bm{m},\bm{k}]\longrightarrow\exp\left[i\frac{2\pi\ell
N_{\chi}T_{\chi}Q}{4}\right]{\cal
Z}[\bm{m},\bm{k}]=\exp\left[i2\pi\ell/2\right]{\cal Z}[\bm{m},\bm{k}]\,,$ (72)

where $N_{\chi}=5$ is the number of the $\chi$ flavors and $T_{\chi}=4$ is the
Dynkin index of $\chi$. Therefore, we expect that $\mathbb{Z}_{4}^{\chi}$
becomes noninvertible in the $SU(6)/\mathbb{Z}_{2}$ chiral theory. Using (50),
the noninvertible operator corresponding to a chiral transformation in
$SU(6)/\mathbb{Z}_{2}$ theory is

$\tilde{U}_{\mathbb{Z}_{4},\ell}=U_{\mathbb{Z}_{4},\ell}\sum_{l_{x}\in\mathbb{Z}}\delta\left(\frac{n_{x}\ell}{2}-l_{x}\right)\sum_{l_{y}\in\mathbb{Z}}\delta\left(\frac{n_{y}\ell}{2}-l_{y}\right)\sum_{l_{z}\in\mathbb{Z}}\delta\left(\frac{n_{z}\ell}{2}-l_{z}\right)\,.$
(73)

Hence, for $\ell\in\\{1,3\\}$, the operator $\tilde{U}_{\mathbb{Z}_{4},\ell}$
projects onto untwisted gauge sectors, and the chiral symmetry operator
becomes noninvertible.

##  4 $SU(N)\times U(1)/\mathbb{Z}_{p}$,
$\mathbb{Z}_{p}\subseteq\mathbb{Z}_{N}$ theories, noninvertible symmetries,
and their anomalies

In this section, we also gauge the $U(1)$ baryon number symmetry. Thus, we are
discussing $SU(N)\times U(1)$ gauge theory with a Dirac fermion in a
representation $\cal R$, $N$-ality $n$, and $U(1)$ charge $+1$. This theory,
as we discussed in Section 2, is endowed with an invertible
$\mathbb{Z}^{\chi}_{\scriptsize 2\mbox{gcd}(T_{\cal R},d_{\cal R})}$ chiral
symmetry as well as an electric $\mathbb{Z}_{N}^{(1)}$ center symmetry acting
on its Wilson’s lines; see Eqs. (17). However, in Anber:2023pny , it was shown
that $SU(N)\times U(1)$ theories also have noninvertible
$\tilde{\mathbb{Z}}^{\chi}_{2T_{\cal R}}$ chiral symmetry. In the following,
we first review the construction of the noninvertible
$\tilde{\mathbb{Z}}^{\chi}_{2T_{\cal R}}$ operator in $SU(N)\times U(1)$
theories, and next, we discuss this operator in $SU(N)\times
U(1)/\mathbb{Z}_{p}$, $\mathbb{Z}_{p}\subseteq\mathbb{Z}_{N}$, theories.

###  4.1 $SU(N)\times U(1)$

Our starting point is the $SU(N)\times U(1)$ theory and its
$\mathbb{Z}^{\chi}_{2T_{\cal R}}$ operator $U_{\mathbb{Z}_{2T_{\cal
R}},\ell}=e^{i\frac{2\pi\ell}{2T_{\cal R}}Q_{5}}$, where $Q_{5}$ is the
conserved chiral charge defined in Eq. (43) in the background of the
fractional $n_{x,y,z}$ and integer $N_{x,y,z}$ magnetic fluxes in the $x,y,z$
directions. We remind that we can turn on fractional fluxes in
$\mathbb{Z}_{N}$ irrespective of the $N$-ality of the matter content since we
use $U(1)$ transition functions to impose the cocycle condition; see Eq.
(2.2). No nontrivial electric twists are applied at this stage, i.e., we take
$\bm{k}\in(N\mathbb{Z})^{3}$, since our nonabelian gauge group is $SU(N)$
rather than $SU(N)/\mathbb{Z}_{p}$. The operator $U_{\mathbb{Z}_{2T_{\cal
R}},\ell}$ is invariant under $SU(N)$. To see that, we apply a large $SU(N)$
gauge transformation, recalling Eq. (24) and setting
$\bm{k}\in(N\mathbb{Z})^{3}$, which immediately gives the change in the
nonabelian winding number by $Q\in\mathbb{Z}$. In addition,
$U_{\mathbb{Z}_{2T_{\cal R}},\ell}$ must be invariant under $U(1)$ gauge
symmetry. The photon gauge field $a_{i}$ transforms under $U(1)$ gauge
symmetry as $a_{j}(x+\hat{e}_{k}L_{k})=a_{j}-\partial_{k}\xi(x)$, and $\xi(x)$
is a periodic gauge function: $\xi(x+\hat{e}_{k}L_{k})=\xi(x)+2\pi p$,
$p\in\mathbb{Z}$. Applying a large $U(1)$ gauge transformation to $Q_{5}$, we
find (see Anber:2023pny  for the derivation)

$\displaystyle U_{\mathbb{Z}_{2T_{\cal R}},\ell}\longrightarrow
U_{\mathbb{Z}_{2T_{\cal R}},\ell}e^{-2\pi
i\ell\left(p_{x}\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{x}+\frac{nn_{x}}{N}\right)+p_{y}\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{y}+\frac{nn_{y}}{N}\right)+p_{z}\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{z}+\frac{nn_{z}}{N}\right)\right)}\,,$
(74)

where $p_{x,y,z}$ are arbitrary integers corresponding to the $U(1)$ gauge
transformation. Eq. (74) shows that the operator $U_{\mathbb{Z}_{2T_{\cal
R}},\ell}$ fails to be gauge invariant under $U(1)$ gauge symmetry. To remedy
this problem, we follow the procedure of the previous section and define a new
operator $\tilde{U}_{\mathbb{Z}_{2T_{\cal R}},\ell}$ by summing arbitrary
copies of the gauge-transformed $U_{\mathbb{Z}_{2T_{\cal R}},\ell}$:

$\displaystyle\tilde{U}_{\mathbb{Z}_{2T_{\cal R}},\ell}$ $\displaystyle=$
$\displaystyle U_{\mathbb{Z}_{2T_{\cal
R}},\ell}\sum_{p_{x},p_{y},p_{z}\in\mathbb{Z}}e^{-2\pi
i\ell\left(p_{x}\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{x}+\frac{nn_{x}}{N}\right)+p_{y}\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{y}+\frac{nn_{y}}{N}\right)+p_{z}\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{z}+\frac{nn_{z}}{N}\right)\right)}$
(75) $\displaystyle=$ $\displaystyle U_{\mathbb{Z}_{2T_{\cal
R}},\ell}\sum_{l_{x}\in\mathbb{Z}}\delta\left(\ell\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{x}+\frac{nn_{x}}{N}\right)-l_{x}\right)\left(\sum_{l_{y}\in\mathbb{Z}}...\right)\left(\sum_{l_{z}\in\mathbb{Z}}...\right)\,.$

The operator $\tilde{U}_{\mathbb{Z}_{2T_{\cal R}},\ell}$ implements the chiral
transformation of the now-noninvertible $\tilde{\mathbb{Z}}^{\chi}_{2T_{\cal
R}}$ symmetry, as it acts projectively by selecting certain nonvanishing
sectors in Hilbert space labeled by the integers $l_{x,y,z}$, such that for
$\ell=1$ we must have

$\displaystyle
l_{x}=\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{x}+\frac{nn_{x}}{N}\right)\in\mathbb{Z}\,,$
(76)

with identical expressions for $l_{y}$ and $l_{z}$. Condition (76) ensures
that all the symmetry elements $\ell=1,2,..,T_{\cal R}$ act invertibly on the
same admissible sector. To explicitly see the projective nature of
$\tilde{U}_{\mathbb{Z}_{2T_{\cal R}},\ell}$ on states in Hilbert space, we use
the partition function of the $SU(N)\times U(1)$ theory given by Eq.
(LABEL:the_partition_function_in_SUNXU1) (we set the electric flux background
$\bm{k}$=0 and, as usual, we use $\bm{n}$ to label a specific fractional
magnetic flux background: $\bm{n}=(n_{x},n_{y},n_{z})$) to compute
$\langle\tilde{U}_{\mathbb{Z}_{2T_{\cal R}},\ell}\rangle$:161616Recall from
our earlier analysis that the theory is endowed with electric
$\mathbb{Z}_{N}^{(1)}$ and magnetic $U^{(1)}_{m}(1)$ symmetries, and the
states of the theory are labeled by the eigenstates of these symmetries,
$\bm{e}$ and $\bm{N}$, respectively.

$\displaystyle\langle\tilde{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}\rangle=\sum_{\bm{e}\in\mathbb{Z}_{N}^{3},\,\bm{N}\in\mathbb{Z}^{3}}{}_{\bm{n}}\langle
E,\bm{e},\bm{N}|e^{-L_{4}\hat{H}}\tilde{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}|E,\bm{e},\bm{N}\rangle_{\bm{n}}\,.$ (77)

We immediately see from the Kronecker deltas in Eq. (75) that only those
sectors with $\bm{N}$ satisfying Eq. (76) are selected.

Turning off the fractional magnetic flux background (i.e., setting
$\bm{n}=0$), the operator $\tilde{U}_{\mathbb{Z}_{2T_{\cal R}},\ell}$ becomes
invertible for $\ell\in T_{\cal R}\mathbb{Z}/\mbox{gcd}(T_{\cal R},d_{\cal
R})$. We recognize that we have just recovered the invertible
$\mathbb{Z}_{\scriptsize 2\mbox{gcd}(T_{\cal R},d_{\cal R})}^{\chi}$ subgroup
of $\tilde{\mathbb{Z}}^{\chi}_{2T_{\cal R}}$. Furthermore, setting $\bm{n}=0$,
the operator $\tilde{U}_{\mathbb{Z}_{2T_{\cal R}},\ell=1}$ destructs all
Hilbert space sectors characterized with integral magnetic fluxes
$\bm{N}\notin T_{\cal R}\mathbb{Z}^{3}/\mbox{gcd}(T_{\cal R},d_{\cal R})$.
This noninvertible nature of the chiral operator should have been anticipated.
When we start with the SU(N) theory with matter, we find an ’t Hooft anomaly
between its invertible $\mathbb{Z}_{2T_{\cal R}}^{\chi}$ chiral symmetry and
$U(1)$ baryon symmetry. This anomaly is valued in $\mathbb{Z}_{\scriptsize
T_{\cal R}/\mbox{gcd}(T_{\cal R},d_{\cal R})}$. Upon gauging $U(1)$, this
anomaly becomes of the ABJ type, and the chiral symmetry becomes
noninvertible. Now, If we take the Euclidean version of our theory in the
infinite volume limit and apply a $\pi/2$ rotation to
$\tilde{U}_{\mathbb{Z}_{2T_{\cal R}},\ell=1}$, the operator becomes a defect.
Alternatively, we may also use the half-gauging procedure to construct this
defect, which was done in Anber:2023pny . Inserting this defect at some
position will generally create a domain wall (since it enforces a chiral
transformation) dressed with a TQFT that accounts for the noninvertible nature
of the defect. It will be interesting to analyze what happens to the domain
walls when we turn on an external magnetic field with flux $\bm{N}\notin
T_{\cal R}\mathbb{Z}^{3}/\mbox{gcd}(T_{\cal R},d_{\cal R})$.

$SU(N)\times U(1)$ gauge theory has an electric $\mathbb{Z}_{N}^{(1)}$
$1$-form global center symmetry, and the immediate exercise would be checking
whether there is a mixed anomaly between the center and the noninvertible
chiral symmetries. To this end, we turn on both electric and magnetic
twists171717Notice that these electric twists $\bm{k}\in\mathbb{Z}^{3}$ are
$\bm{e}$ that label the physical states in Hilbert space:
$|E,\bm{e},\bm{N}\rangle_{\bm{n}}$ . In principle, $k_{j}$ should be in
$\mathbb{Z}$ Mod $N$, but, as usual, we drop the modding as this does not
affect the cocycle conditions. $(\bm{m},\bm{k})\in\mathbb{\mathbb{Z}}^{6}$,
giving rise to nonabelian fractional topological charge
$Q_{SU(N)}\in\mathbb{Z}/N$ as well as abelian topological charge
$Q_{u}=\left(\frac{n}{N}\right)^{2}$; see Eq. (15). Using Eqs. (24, 44),
setting $\bm{k}=(1,0,0)$, we find

$\displaystyle T_{x}t_{x}\tilde{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}(T_{x}t_{x})^{-1}$ $\displaystyle=$
$\displaystyle\tilde{U}_{\mathbb{Z}_{2T_{\cal R}},\ell}e^{-2\pi
i\ell\left(\frac{n_{x}}{N}-\frac{n}{N}\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{x}+\frac{nn_{x}}{N}\right)\right)}$
(78) $\displaystyle=$ $\displaystyle\tilde{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}e^{-i2\pi\ell\left(\frac{n_{x}-nl_{x}}{N}\right)}\,,$

and we used Condition (76) to go from the first to the second line. If the
phase is nontrivial, then there is a mixed anomaly between the electric
$\mathbb{Z}_{N}^{(1)}$ $1$-form center and the $0$-form noninvertible
$\tilde{\mathbb{Z}}^{\chi}_{2T_{\cal R}}$ symmetries, leading to spectral
degeneracy of states (those that already selected by the operator
$\tilde{U}_{\mathbb{Z}_{2T_{\cal R}},\ell}$). The algebra defined by the
commutation relations $[H,T_{j}t_{j}]=[H,\tilde{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}]=0$ along with the mixed anomaly (78), under the assumption of a
nontrivial phase, furnishes a finite-dimensional space with dimension
$N/\mbox{gcd}(N,n_{x}-nl_{x})$ (we take $n_{x}=n_{y}=n_{z}$). The Hilbert
space of physical states, which are labeled by $N$ different electric fluxes
$\bm{e}$, sit in $N/\mbox{gcd}(N,n_{x}-nl_{x})$ orbits, and a rotation by
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell=1}$ links a state with a flux
$e_{j}$ to a state with a flux $e_{\scriptsize j}+\mbox{gcd}(N,n_{j}-nl_{j})$,
i.e., they have the same energy.

###  4.2 $SU(N)\times U(1)/\mathbb{Z}_{p}$,
$\mathbb{Z}_{p}\subseteq\mathbb{Z}_{N}$

Next, we study the noninvertible operators in $SU(N)\times
U(1)/\mathbb{Z}_{p}$ gauge theory, where $\mathbb{Z}_{p}$ is a subgroup of the
$\mathbb{Z}_{N}$ center symmetry. This theory has an electric
$\mathbb{Z}_{N/p}^{(1)}$ $1$-form global symmetry acting on the $p$-th power
of the spatial components of the abelian and nonabelian Wilson’s lines defined
in Eq. (17):

$\displaystyle W_{j,\,SU(N)}^{e_{j}p}\,,W_{j,\,U(1)}^{e_{j}p}\,,\quad
e_{j}=1,2,..,N/p,\quad j=1,2,3\,.$ (79)

These Wilson’s lines are invariant under $\mathbb{Z}_{p}$, as they should be,
as this symmetry is gauged. Notice that the allowed abelian probe charges $q$
need to satisfy $q=z_{e}$, where $z_{e}=e_{j}p$ is the $N$-ality of the
nonabelian line. Thus, we can represent the lines in Eq. (79) by the pair
$(z_{e},q=z_{e})$. The theory also possesses a magnetic $U^{(1)}_{m}(1)$
$1$-form symmetry acting on ’t Hooft lines. Let $z_{m}=0,1,..,p-1$, and $g$ be
the $N$-ality of the nonabelian ’t Hooft line and the abelian magnetic charge,
respectively. Then, the pairs $(z_{e},q=z_{e})$ and $(z_{m},g)$ must satisfy
the Dirac quantization condition $e^{i2\pi(-qg+z_{e}z_{m}/p)}=1$ or
$z_{e}z_{m}-pqg\in p\mathbb{Z}$, which gives a constraint on the magnetic
charges: $g=\frac{z_{m}}{p}+\mathbb{Z}$, i.e., the abelian magnetic charges
can be fractional Tong:2017oea . Another way of putting it is that the
presence of the Abelian Wilson’s lines $W_{j,\,U(1)}^{e_{j}p}$ demand that the
Abelian ’t Hooft lines are ${\cal T}_{j,\,U(1)}^{N_{j}+n_{j}/p}$,
$n_{j}\in\mathbb{Z}_{p}$, $N_{j}\in\mathbb{Z}$, such that the electric and
magnetic lines are mutually local. The physical states in Hilbert space are
taken to be eigenstates of the commuting set of the Hamiltonian, the
generators of electric symmetry, and the generators of magnetic symmetry:

$\displaystyle|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}$ $\displaystyle=$
$\displaystyle|E,p\bm{e},\bm{n}/p+\bm{N}\rangle_{\bm{m}}\,,\quad
e_{j}=0,1,..,N/p-1\,,\quad N_{j}\in\mathbb{Z}\,\quad n_{j}=0,1,..,p-1\,,$ (80)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad j=1,2,3\,,$

and $\bm{m}\in\mathbb{Z}^{3}$ is the fractional magnetic flux background (or
background magnetic twist). Remember that, in principle,
$\bm{m}\in(\mathbb{Z}\,\mbox{Mod}\,N)^{3}$; however, we drop the modding by
$N$ since this cannot affect the cocycle condition. Notice that we can always
activate a $\mathbb{Z}_{N}$ magnetic twist since, as emphasized several times,
we use a combination of nonabelian and abelian transition functions. Also, in
the special case $p=N$, we should remove the subscript $\bm{m}$ since, in this
case, the Hilbert space is spanned by eigenstates of the full magnetic
$\mathbb{Z}_{N}$ fluxes, i.e., $n_{j}=0,1,...,N-1$.

The operator $\tilde{U}_{\mathbb{Z}_{2T_{\cal R}},\ell}$ defined in Eq. (75)
is invariant under both $SU(N)$ and $U(1)$ gauge transformations. However,
because we are now gauging $\mathbb{Z}_{p}$, the operator must also be
invariant under $\mathbb{Z}_{p}$ gauge transformations. Let us recall that
$T_{j}t_{j}$ is the generator of the electric $\mathbb{Z}_{N}^{(1)}$ symmetry,
and therefore, $\left(T_{j}t_{j}\right)^{N/p}$ generates the $\mathbb{Z}_{p}$
symmetry, which must be gauged. The action of $\left(T_{j}t_{j}\right)^{N/p}$
on $\tilde{U}_{\mathbb{Z}_{2T_{\cal R}},\ell}$ can be read from the first line
in Eq. (78) by applying the operation $N/p$ times:

$\displaystyle\left(T_{j}t_{j}\right)^{N/p}\tilde{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}\left(T_{j}t_{j}\right)^{-N/p}=\tilde{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}e^{-2\pi
i\ell\left(\frac{n_{x}}{p}-\frac{n}{p}\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{x}+\frac{nn_{x}}{N}\right)\right)}\,.$
(81)

This relation shows that for a general $\ell$,
$\tilde{U}_{\mathbb{Z}_{2T_{\cal R}},\ell}$ fails to be gauge invariant under
a $\mathbb{Z}_{p}$ gauge transformation181818In the special case $p=1$, the
phase becomes $e^{2\pi i\ell
n\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{x}+\frac{nn_{x}}{N}\right)}$,
and using Condition (76), the phase trivializes. This shows that this operator
is gauge invariant in $SU(N)\times U(1)$ theory, as expected.. Being
acquainted with the remedy of this problem, we use
$\left(T_{j}t_{j}\right)^{N/p}\tilde{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}\left(T_{j}t_{j}\right)^{-N/p}$ as a building block of a gauge
invariant operator by summing over arbitrary copies of the block. The
noninvertible operator is then given by

$\displaystyle\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$
$\displaystyle=\sum_{p_{x},p_{y},p_{z}\in\mathbb{Z}}(T_{x}t_{x})^{\frac{Np_{x}}{p}}(T_{y}t_{y})^{\frac{Np_{y}}{p}}(T_{z}t_{z})^{\frac{Np_{z}}{p}}U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}(T_{x}t_{x})^{-\frac{Np_{x}}{p}}(T_{y}t_{y})^{-\frac{Np_{y}}{p}}(T_{z}t_{z})^{-\frac{Np_{z}}{p}}$
$\displaystyle=U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\sum_{p_{x},p_{y},p_{z}\in\mathbb{Z}}e^{-2\pi
i\ell\left(\frac{n_{x}}{p}-\frac{n}{p}\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{x}+\frac{nn_{x}}{N}\right)\right)+(x\rightarrow
y)+(x\rightarrow z)}$
$\displaystyle=U_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}\sum_{l_{x}\in\mathbb{Z}}\delta\left(\frac{\ell
n_{x}}{p}-\frac{\ell
n}{p}\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{x}+\frac{nn_{x}}{N}\right)-l_{x}\right)\left(\sum_{l_{y}\in\mathbb{Z}}...\right)\left(\sum_{l_{z}\in\mathbb{Z}}...\right)\,.$
(82)

The operator $\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$ acts invertibly
on sectors in Hilbert space that, for $\ell=1$, satisy the condition

$\displaystyle
l_{x}=\frac{n_{x}}{p}-\frac{n}{p}\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{x}+\frac{nn_{x}}{N}\right)\in\mathbb{Z}\,,$
(83)

with identical expressions in the $y$ and $z$ directions. The operator
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$, as introduced in Eq. (82),
within the context of $SU(N)\times U(1)/\mathbb{Z}_{p}$ gauge theory, is a
generalization of the operator defined in Eq. (75) for the conventional
$SU(N)\times U(1)$ theory. Furthermore, Condition (83) represents a broader
generalization of Condition (76). In the specific scenario where $p=N$ holds,
corresponding to the $SU(N)\times U(1)/\mathbb{Z}_{N}$ theory, Condition (83)
precisely mirrors the criterion for the absence of a mixed anomaly between the
electric $\mathbb{Z}_{N}^{(1)}$ $1$-form global symmetry and the noninvertible
chiral symmetry inherent to the $SU(N)\times U(1)$ theory. This correspondence
is clear from the first line of Eq. (78).

The $SU(N)\times U(1)/\mathbb{Z}_{p}$ theory exhibits an electric
$\mathbb{Z}_{N/p}^{(1)}$ one-form global symmetry, which is generated by the
operators $(T_{j}t_{j})^{p}$. When introducing a background for this symmetry,
we uncover a mixed anomaly between the noninvertible chiral symmetry and the
$\mathbb{Z}_{N/p}^{(1)}$ symmetry. Sandwiching
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}$, defined in Eq. (82), between
$(T_{x}t_{x})^{p}$ and $(T_{x}t_{x})^{-p}$ and using Eqs. (24, 44), we find

$\displaystyle(T_{x}t_{x})^{p}\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell}(T_{x}t_{x})^{-p}$
$\displaystyle=$ $\displaystyle\tilde{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}e^{-2\pi
i\ell\left(\frac{pn_{x}}{N}-\frac{pn}{N}\frac{d_{\mathcal{R}}}{T_{\mathcal{R}}}\left(N_{x}+\frac{nn_{x}}{N}\right)\right)}$
(84) $\displaystyle=$ $\displaystyle\tilde{U}_{\mathbb{Z}_{2T_{\cal
R}},\ell}e^{-i2\pi l_{x}\ell\frac{p^{2}}{N}}\,,$

where we used $l_{x}$ defined in Eq. (83) in going from the first to the
second line. When the phase $e^{-i2\pi l_{x}\ell\frac{p^{2}}{N}}$ is
nontrivial, it signifies the presence of a degeneracy within the spectrum.
Notice that the anomaly phase coincides with the phase in Eq. (63) if we set
$q=N$ in the latter. This should not surprise us since, in this section, we
employ the full $\mathbb{Z}_{N}$ center symmetry, thanks to gauging $U(1)$.
The anomaly in (84) is valued in $\mathbb{Z}_{\scriptsize
N/\mbox{gcd}(N,p^{2}l_{x})}$ (we take $n_{x}=n_{y}=n_{z}$) indicating a
$N/\mbox{gcd}(N,p^{2}l_{x})$-fold degeneracy. The Hilbert space of physical
states, which are labeled by $N/p$ distinct electric fluxes, sit in
$N/\mbox{gcd}(N,p^{2}l_{x})$ orbits, and a rotation by
$\tilde{U}_{\mathbb{Z}_{2T_{\mathcal{R}}},\ell=1}$ maps a state with an
electric flux $pe_{j}$ to a state with a flux $p(e_{\scriptsize
j}+\mbox{gcd}(N,p^{2}l_{j})/p)$, i.e., they have the same energy.

###  4.3 Examples

####  4.3.1 $SU(4k+2)\times U(1)/\mathbb{Z}_{p}$ with $2$-index antisymmetric
fermions

$SU(4k+2)\times U(1)$ theory with a single $2$-index anti-symmetric Dirac
fermion was considered in Anber:2023pny . Here, we study this theory when we
gauge a $\mathbb{Z}_{p}\subseteq\mathbb{Z}_{N}$ subgroup of the center.
Numerical scans reveal that condition (83) is always satisfied for specific
values of $n_{x}$ and $N_{x}$. Also, the anomaly (84) is trivial unless both
$p$ and $l_{x}$ are odd; then, the anomaly is valued in $\mathbb{Z}_{2}$. The
Hilbert space is spanned by the physical states

$\displaystyle|\psi\rangle_{\scriptsize\mbox{phy}\,,\bm{m}}$ $\displaystyle=$
$\displaystyle|E,p\bm{e},\bm{n}/p+\bm{N}\rangle_{\bm{m}}\,,$
$\displaystyle\quad e_{j}=0,1,..,(4k+2)/p-1\,,\quad N_{j}\in\mathbb{Z}\,,\quad
n_{j}=0,1,..,p-1\,,\quad j=1,2,3\,,$

and the anomaly means that the states live in two orbits such that
$|E,p\bm{e},\bm{n}/p+\bm{N}\rangle_{\bm{m}}$,
$|E,p(\bm{e}+\mbox{gcd}(N,p^{2}\bm{l})/p),\bm{n}/p+\bm{N}\rangle_{\bm{m}}$,
$|E,p(\bm{e}+2\mbox{gcd}(N,p^{2}\bm{l})/p),\bm{n}/p+\bm{N}\rangle_{\bm{m}}$,
etc. have the same energy (we take $n_{x}=n_{y}=n_{z}$).

####  4.3.2 The Standard Model

The methods presented in this paper provide a systematic means to find
noninvertible symmetries in any given gauge theory. As an important
application, we employ our approach to search for noninvertible symmetries in
the nongravitational sector of the Standard Model (SM). SM is based on
$su(3)\times su(2)\times u(1)$ Lie algebra. Yet, the faithful gauge group,
i.e., the global structure of the group, is to be uncovered. The matter
content and charges under the gauge and global symmetries are displayed in
Table 1, and all fermions are taken to be left-handed Weyls.

$\begin{array}[]{|c|ccc|cc|}\hline\cr\text{field}&SU(3)&SU(2)&U(1)&U(1)_{B}&U(1)_{L}\cr\hline\cr
q_{L}&\square&\square&1&{1\over 3}&0\cr
l_{L}&\textbf{1}&\square&-3&0&1\cr\tilde{e}_{R}&\textbf{1}&\textbf{1}&6&0&-1\cr\tilde{u}_{R}&\overline{\square}&\textbf{1}&-4&-{1\over
3}&0\cr\tilde{d}_{R}&\overline{\square}&\textbf{1}&2&-{1\over 3}&0\cr
h&\textbf{1}&\square&3&0&0\cr\hline\cr\end{array}$ Table 1: Matter content and
charges of SM: $q_{L}$ and $l_{L}$ are the quark and lepton doublets,
$\tilde{e}_{R},\tilde{u}_{R},\tilde{d}_{R}$ are the electron and up and down
quarks singlets, while $h$ is the Higgs doublet. Notice that we take the hyper
$U(1)$ charges to be integers, while the matter content has the standard
charges under the baryon number $U(1)_{B}$ and lepton number $U(1)_{L}$
symmetries.

The anomalies associated with the $U(1)_{B}$ and $U(1)_{L}$ symmetries are
given by: $U(1)_{B}\times[SU(2)]^{2}=U(1)_{L}\times[SU(2)]^{2}=1$,
$U(1)_{B}\times[SU(3)]^{2}=U(1)_{L}\times[SU(3)]^{2}=0$,
$U(1)_{B}\times[U(1)]^{2}=U(1)_{L}\times[U(1)]^{2}=-18$ . Thus, we see that
$U(1)_{B-L}$ symmetry is anomaly-free symmetry (we neglect gravity in this
context). Under a $U(1)_{B+L}$ rotation, the path integral picks up an ABJ
phase

$\exp\left(i\alpha\cdot N_{f}(2c_{2}(F)-36c_{2}(f))\right)\,,$ (86)

where $N_{f}$ is the number of families, $c_{2}(F)$ is the second Chern class
for $SU(2)$ and $c_{2}(f)$ is the second Chern class for $U(1)$. The ABJ
anomaly breaks the $U(1)_{B+L}$ down to a
$\mathbb{Z}^{B+L}_{\scriptsize\mbox{gcd}(2,36)N_{f}}=\mathbb{Z}^{B+L}_{2N_{f}}$
symmetry. Notice that $SU(3)$ does not play a role in the ABJ anomaly.

The matter content is consistent with the existence of an electric
$\mathbb{Z}_{6}^{(1)}$ $1$-form global symmetry Tong:2017oea ; Anber:2021upc .
The cocycle conditions satisfied by SM on $\mathbb{T}^{4}$ with a gauged
$\mathbb{Z}_{6}^{(1)}$ are given by Anber:2021upc :

$\displaystyle\Omega_{(3)\mu}(x^{\nu}=L^{\nu})\Omega_{(3)\nu}(x^{\mu}=0)$
$\displaystyle=e^{2\pi
i\frac{n^{(3)}_{\mu\nu}}{3}}\Omega_{(3)\nu}(x^{\mu}=L^{\mu})\Omega_{(3)\mu}(x^{\nu}=0)\,,$
$\displaystyle\Omega_{(2)\mu}(x^{\nu}=L^{\nu})\Omega_{(2)\nu}(x^{\mu}=0)$
$\displaystyle=e^{2\pi
i\frac{n^{(2)}_{\mu\nu}}{2}}\Omega_{(2)\nu}(x^{\mu}=L^{\mu})\Omega_{(2)\mu}(x^{\nu}=0)\,,$
(87) $\displaystyle\omega_{(1)\mu}(x^{\nu}=L^{\nu})\omega_{(1)\nu}(x^{\mu}=0)$
$\displaystyle=e^{-2\pi
i(\frac{n^{(3)}_{\mu\nu}}{3}+\frac{n^{(2)}_{\mu\nu}}{2})}\omega_{(1)\nu}(x^{\mu}=L^{\mu})\omega_{(1)\mu}(x^{\nu}=0)\,.$

$\Omega_{(i)}$, $i=2,3$, and $\omega_{(1)}$ are the transition functions of
the gauge bundles, $n^{(i)}_{\mu\nu}$ are the ’t Hooft twists, and the
superscript/subscript $(i)=(3),(2),(1)$ denote the condition for the
$SU(3),SU(2),U(1)$ gauge groups respectively. The electric
$\mathbb{Z}_{6}^{(1)}$ symmetry is generated by a combinations of the $SU(3)$
center, $T^{(3)}_{j}$, the $SU(2)$ center, $T^{(2)}_{j}$, and the $U(1)$
center $t_{j}$, such that the full $\mathbb{Z}_{6}^{(1)}$ symmetry generator
is given by $T^{(3)}_{j}T^{(2)}_{j}t_{j}$, $j=x,y,z$.

The anomalous $U(1)_{B+L}$ current conservation law is given by

$\partial_{\mu}j^{\mu}_{B+L}-2N_{f}\partial_{\mu}K^{\mu}_{SU(2)}(A)+\frac{36N_{f}}{8\pi^{2}}\epsilon_{\mu\nu\lambda\sigma}\partial^{\mu}a^{\nu}\partial^{\lambda}a^{\sigma}=0\,,$
(88)

where $K^{\mu}_{SU(2)}$ is the $SU(2)$ topological current. The corresponding
unbroekn $\mathbb{Z}^{B+L}_{2N_{f}}$ symmetry operator on $\mathbb{T}^{3}$ is
given by:

$\displaystyle
U_{\mathbb{Z}_{2N_{f}},\ell}=\exp\left[i\frac{2\pi\ell}{2N_{f}}Q_{5}\right]\,,$
(89)

where the conserved charge $Q_{5}$ is given by (here we turn on a
$\mathbb{Z}_{6}$ magnetic twist)

$\displaystyle Q_{5}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}^{3}}d^{3}x\left[j_{B+L}^{0}-2N_{f}K^{0}_{SU(2)}(A)+\frac{36N_{f}}{8\pi^{2}}\epsilon^{ijk}a_{i}\partial_{j}a_{k}\right]$
(90) $\displaystyle-{18N_{f}\over 4\pi}(N_{z}+{1\over
6}n_{z})\left[\int\limits_{0}^{L_{y}}{dy\over
L_{y}}\int\limits_{0}^{L_{z}}dza_{z}(x=0,y,z)+\int\limits_{0}^{L_{x}}{dx\over
L_{x}}\int\limits_{0}^{L_{z}}dza_{z}(x,y=0,z)\right]$
$\displaystyle+\sum\limits_{\scriptsize\mbox{cyclic}}(x\rightarrow
y\rightarrow z\rightarrow x)\,.$

Under a $U(1)$ gauge transformation, $U_{\mathbb{Z}_{2N_{f}},\ell}$ transforms
as

$\displaystyle U_{\mathbb{Z}_{2N_{f}},\ell}\longrightarrow
U_{\mathbb{Z}_{2N_{f}},\ell}e^{-i2\pi\left(\frac{18\ell
N_{f}}{N_{f}}\left(N_{x}+\frac{n_{x}}{6}\right)\right)+(x\rightarrow
y)+(x\rightarrow z)}=U_{\mathbb{Z}_{2N_{f}},\ell}\,.$ (91)

Therefore, $U_{\mathbb{Z}_{2N_{f}},\ell}$ is $U(1)$ gauge invariant, as
required. Further, we examine $U_{\mathbb{Z}_{2N_{f}},\ell}$ after gauging the
electric $\mathbb{Z}_{6}^{(1)}$ $1$-form center by sandwiching
$U_{\mathbb{Z}_{2N_{f}},\ell}$ between its generators (this is a
generalization of Eq. (78)):

$\displaystyle
T_{x}^{(3)}T_{x}^{(2)}t_{x}U_{\mathbb{Z}_{2N_{f}},\ell}\left(T_{x}^{(3)}T_{x}^{(2)}t_{x}\right)^{-1}$
$\displaystyle=$
$\displaystyle\underbrace{e^{-i\frac{2\pi\ell(2N_{f})}{2N_{f}}\frac{n_{x}^{(2)}}{2}}}_{\scriptsize\mbox{from}\,K^{0}_{SU(2)}(A)}\underbrace{e^{i2\pi\ell\frac{36N_{f}}{2N_{f}}\left(\frac{1}{6}\right)\left(N_{x}+\frac{n_{x}^{(2)}}{2}+\frac{n_{x}^{(3)}}{3}\right)}}_{\scriptsize\mbox{from}\,\epsilon^{ijk}a_{i}\partial_{j}a_{k}}U_{\mathbb{Z}_{2N_{f}},\ell}$
(92) $\displaystyle=$ $\displaystyle U_{\mathbb{Z}_{2N_{f}},\ell}\,.$

We used Eq. (33), setting $k_{x}=m_{x}=1$, to find the first exponent. The
second exponent is found by applying Eq. (44) and using $n=1$, $N=6$. Here,
$n_{x}^{(2)}$, $n_{x}^{(3)}$, and $N_{x}$ are the $SU(2)$ and $SU(3)$
fractional twists and $U(1)$ integral magnetic flux, respectively. This
analysis shows that SM does not possess noninvertible symmetries in its
nongravitational sectors. Our findings are consistent with Putrov:2023jqi .

##  5 Coupling gauge theories to axions and noninvertible symmetries

In this section, we introduce axions into the game, taking $\mathbb{T}^{4}$ to
be larger than any scale in the theory. To be specific, we take
$SU(N)/\mathbb{Z}_{p}$ or $SU(N)\times U(1)/\mathbb{Z}_{p}$ gauge theories of
the previous sections and follow the setup of Anber:2020xfk  by adding a
complex scalar $\Phi$ that is neutral under the gauge groups but couples to
the fermions. Thus, we add the following terms to the Lagrangian:

$\displaystyle{\cal
L}\supset|\partial_{\mu}\Phi|^{2}+V(\Phi)+y\Phi\tilde{\psi}\psi+\mbox{h.c.}\,,$
(93)

where $\psi,\tilde{\psi}$ are two left-handed Weyl fermions in representations
${\cal R}$ and its complex conjugate $\bar{\cal R}$, respectively, and $y$ is
a Yukawa coupling. The potential of the complex field is
$V(\Phi)=\lambda(|\Phi|^{2}-v^{2})^{2}$, where $\lambda$ is ${\cal O}(1)$
dimensionless parameter. We take the scalar field v.e.v. $v\gg\Lambda$, where
$\Lambda$ the strong scale of the gauge sector. We shall pretend that we did
not know about the noninvertible symmetries or how to construct them, and let
us see if we can reproduce them in the IR.

Let us first consider the $SU(N)$ gauge theory before gauging $U(1)$ and the
electric $\mathbb{Z}_{p}^{(1)}$ symmetry. Under $\mathbb{Z}_{2T_{\cal
R}}^{\chi}$ and $U(1)$ baryon number, the different fields transform as

$\displaystyle\mathbb{Z}_{2T_{\cal R}}^{\chi}$ $\displaystyle:$
$\displaystyle\quad\Phi\longrightarrow e^{i\frac{-2\pi}{T_{\cal
R}}}\Phi\,,\psi\longrightarrow e^{i\frac{2\pi}{2T_{\cal
R}}}\psi\,,\tilde{\psi}\longrightarrow e^{i\frac{2\pi}{2T_{\cal
R}}}\tilde{\psi}\,,$ $\displaystyle U(1)$ $\displaystyle:$
$\displaystyle\quad\Phi\longrightarrow\Phi\,,\psi\longrightarrow
e^{i\alpha}\psi\,,\tilde{\psi}\longrightarrow
e^{-i\alpha}\tilde{\psi}\,,\alpha\in[0,2\pi)\,.$ (94)

If we write $\Phi$ as $\Phi=\rho e^{i\varphi}$, where $\varphi$ is the axion,
then $\varphi$ transforms under $\mathbb{Z}^{\chi}_{2T_{\cal R}}$ as

$\displaystyle\varphi\longrightarrow\varphi-\frac{2\pi}{T_{\cal R}}\,$ (95)

and notice that the axion is inert under the $\mathbb{Z}_{2}^{F}$ fermion
number subgroup of $\mathbb{Z}_{2T_{\cal R}}^{\chi}$ .

Next, we consider $SU(N)/\mathbb{Z}_{p}$ or $SU(N)\times U(1)/\mathbb{Z}_{p}$
gauge theories with axions. Flowing to an energy scale below $v$, the radial
degree of freedom $\rho$ freezes in, i.e., we set $\rho=v$, and the fermions
acquire a mass $\sim yv$ and decouple. What remains is the light degree of
freedom, the axion $\varphi$. However, the axion should reproduce all the UV
anomalies. Thus, we can write the following IR effective Lagrangian of
$\varphi$:

$\displaystyle{\cal
L}_{\varphi}=v^{2}\left(\partial_{\mu}\varphi\right)^{2}+T_{\cal
R}\varphi\frac{\mbox{tr}(F\wedge F)}{8\pi^{2}}+d_{\cal R}\varphi\frac{f\wedge
f}{8\pi^{2}}\,.$ (96)

Variation of ${\cal L}_{\varphi}$ w.r.t $\varphi$ produces the anomalous
current conservation law:

$\displaystyle\partial_{\mu}j^{\mu}_{(\varphi)}-T_{\cal
R}\partial_{\mu}K^{\mu}(A)-\frac{d_{\cal
R}}{8\pi^{2}}\epsilon_{\mu\nu\lambda\sigma}\partial^{\mu}a^{\nu}\partial^{\lambda}a^{\sigma}=0\,,$
(97)

where $j^{\mu}_{(\varphi)}=v^{2}\partial^{\mu}\varphi$. This is exactly the
anomalous current conservation law we had previously, now written down for the
axion current. Therefore, everything we said in the previous sections applies
here. In particular, we can define an operator of the $\mathbb{Z}_{2T_{\cal
R}}^{\chi}$ symmetry as:

$\displaystyle{\cal U}_{\mathbb{Z}_{2T_{\cal
R}}\,,\ell}=\exp\left[i\frac{2\pi\ell}{T_{\cal
R}}\int_{\mathbb{T}^{3}}(j_{(\varphi)}^{0}-T_{\cal R}K^{0}(A))+...\right]\,,$
(98)

where the dots denote the contribution from the $U(1)$ gauge field (see Eq.
(43)). We used a calligraphic letter for the operator to emphasize that it is
constructed in the IR. Yet, all the anomalies and failure of invariance under
gauge symmetries that lead to the noninvertibility of the UV operators apply
here as well. Thus, similar to what we did before, we can construct the
noninvertible operator $\tilde{\cal U}_{\mathbb{Z}_{2T_{\cal R}}\,,\ell}$,
which implements the noninvertible symmetry $\tilde{\mathbb{Z}}_{2T_{\cal
R}}^{\chi}$ in the IR. Such operators shall project onto magnetic sectors and
also exhibit mixed anomalies with the global $1$-form electric center
symmetry, exactly as we discussed previously.

It was pointed out in Cordova:2023her  that $SU(N)/\mathbb{Z}_{p}$ theories
with axions have noninvertible symmetries. However, our construction shows
that such a conclusion is not general and depends on the UV completion.
Consider two theories $SU(4k)/\mathbb{Z}_{2}$ and $SU(4k+2)/\mathbb{Z}_{2}$
with a Dirac fermion in the $2$-index antisymmetric representation and coupled
to a complex scalar field $\Phi$ as above. As we flow to the IR, we can
construct the operators corresponding to the chiral symmetries. We discussed
in Section 3.3.1 that $SU(4k)/\mathbb{Z}_{2}$ theory does not exhibit an
anomaly between its chiral symmetry and the $1$-form symmetry of the
corresponding $SU(4k)$ theory, and hence, the chiral symmetry operator is
invertible. Therefore, an axion domain wall (DW), implemented by the action of
$\tilde{\cal U}_{\mathbb{Z}_{8k-4}\,,\ell}$, will not be dressed with TQFT
degrees of freedom. On the contrary, $SU(4k+2)/\mathbb{Z}_{2}$ exhibits an
anomaly between its chiral symmetry and the $1$-form center of the
corresponding $SU(4k+2)$ theory, and thus, the minimal chiral symmetry
operator $\tilde{\cal U}_{\mathbb{Z}_{8k}\,,\ell=1}$ is noninvertible. The
axion DW implemented by $\tilde{\cal U}_{\mathbb{Z}_{8k}\,,\ell=1}$ must be
dressed with a fractional quantum Hall TQFT.

We may also consider axions in $SU(N)\times U(1)/\mathbb{Z}_{p}$ theory of
Section 4. Everything we said there is transcendent to the IR axion domain
walls. In particular, for $p=1$, the operator $\tilde{\cal
U}_{\mathbb{Z}_{2T_{\cal R}}\,,\ell=1}$ destructs the Hilbert space sectors
characterized by vanishing fractional $\bm{n}=0$ and integral magnetic fluxes
$\bm{N}\notin T_{\cal R}\mathbb{Z}^{3}/\mbox{gcd}(T_{\cal R},d_{\cal R})$. It
will be interesting to examine what happens to the axion domain walls of this
theory as we place them in such an external magnetic field.

Acknowledgments:

We would like to thank Erich Poppitz and Tin Sulejmanpasic for various
illuminating discussions. We also thank Erich Poppitz for comments on the
manuscript. This work is supported by STFC through grant ST/T000708/1.

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