aainstitutetext: Centre for Particle Theory, Department of Mathematical
Sciences, Durham University, South Road, Durham DH1 3LE, UKbbinstitutetext:
Department of Physics, University of Toronto, 60 St George St., Toronto, ON
M5S 1A7, Canada

# Noninvertible anomalies in $SU(N)\times U(1)$ gauge theories

Mohamed M. Anber b Erich Poppitz<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

W

e study $4$-dimensional $SU(N)\times U(1)$ gauge theories with a single
massless Dirac fermion in the $2$-index symmetric/antisymmetric
representations and show that they are endowed with a noninvertible $0$-form
$\tilde{\mathbb{Z}}_{2(N\pm 2)}^{\chi}$ chiral symmetry along with a $1$-form
$\mathbb{Z}_{N}^{(1)}$ center symmetry. By using the Hamiltonian formalism and
putting the theory on a spatial three-torus $\mathbb{T}^{3}$, we construct the
non-unitary gauge invariant operator corresponding to
$\tilde{\mathbb{Z}}_{2(N\pm 2)}^{\chi}$ and find that it acts nontrivially in
sectors of the Hilbert space characterized by selected magnetic fluxes. When
we subject $\mathbb{T}^{3}$ to $\mathbb{Z}_{N}^{(1)}$ twists, for $N$ even, in
selected magnetic flux sectors, the algebra of $\tilde{\mathbb{Z}}_{2(N\pm
2)}^{\chi}$ and $\mathbb{Z}_{N}^{(1)}$ fails to commute by a $\mathbb{Z}_{2}$
phase. We interpret this noncommutativity as a mixed anomaly between the
noninvertible and the $1$-form symmetries. The anomaly implies that all states
in the torus Hilbert space with the selected magnetic fluxes exhibit a two-
fold degeneracy for arbitrary $\mathbb{T}^{3}$ size. The degenerate states are
labeled by discrete electric fluxes and are characterized by nonzero
expectation values of condensates. In an Appendix, we also discuss how to
construct the corresponding noninvertible defect via the “half-space gauging”
of a discrete one-form magnetic symmetry.

##  1 Introduction

Symmetries are the backbone of quantum field theory (QFT); the spectrum of
local and extended operators are organized in symmetry representations. The
modern way to define symmetry is via its action on topological surfaces. A
$p$-form symmetry acts on $p$-dimenstional objects and is generated by
operators supported on $p+1$-codimensional surfaces Gaiotto:2014kfa .
Traditionally, symmetries are connected to groups, and the operator that
generates the symmetry is unitary. However, recent developments have
highlighted the need to broaden the definition of symmetries to include
actions generated by nonunitary operators. These symmetries are noninvertible
since the corresponding operators do not have an inverse. Consequently, these
operations do not form groups but can be comprehended as categories, offering
a new perspective on the nature of symmetries.

Noninvertible symmetries were first identified and applied in $2$-dimensional
QFT, see, e.g., Fuchs:2002cm ; Komargodski:2020mxz . The appreciation of the
role of this new development in $4$-dimensional QFT resulted in an avalanche
of works on this topic (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 ).

One significant development in the field is the recognition that
$4$-dimensional quantum electrodynamics (QED) possesses a noninvertible
symmetry Choi:2022jqy ; Cordova:2022ieu . The idea is that QED with a single
Dirac fermion has a classical $U(1)_{\chi}$ chiral symmetry broken to the
$\mathbb{Z}_{2}$ fermion number by the ABJ anomaly. The Noether current
corresponding to $U(1)_{\chi}$ is not conserved as it receives a contribution
from the anomaly. However, one may define a conserved chiral current by
subtracting a Chern-Simons term that encodes the anomaly. Next, an operator
corresponding to $U(1)_{\chi}$ is constructed by exponentiating the modified
current and integrating it on a $3$-surface. This operator is not gauge
invariant. However, dressing it with a TQFT makes it gauge invariant. The
resulting operator can be shown to generate a noninvertible symmetry for every
rational value of the $U(1)_{\chi}$ parameter. The construction in
Choi:2022jqy ; Cordova:2022ieu  was further developed in Karasik:2022kkq ;
GarciaEtxebarria:2022jky  by showing that the definition of the noninvertible
operator can be extended to any real parameter of $U(1)_{\chi}$ by coupling
the theory to a scalar field living on the $3$-surface.

Comprehending this novel structure in gauge theories is important in pursuing
a deeper understanding of QFT. The present work examines $SU(N)\times U(1)$
gauge theory with a single massless Dirac fermion in a representation $R$. The
theory has a classical $U(1)_{\chi}$ symmetry broken by the ABJ anomaly in
$SU(N)$\- and $U(1)$-instanton backgrounds. Does this theory exhibit
noninvertible symmetries, and if so, can they be utilized to establish exact
nonperturbative statements? We demonstrate that the answer to this query is
affirmative.

Unlike QED, when our theory is put on a general manifold, the chiral symmetry
is reduced to $\mathbb{Z}_{\scriptsize 2\mbox{gcd}(T_{R},d_{R})}^{\chi}$,
where $T_{R}$ and $d_{R}$ are the Dynkin index and dimension of $R$.
Interestingly, we also show that the theory possesses a noninvertible
$\tilde{\mathbb{Z}}_{2T_{R}}^{\chi}$ discrete chiral symmetry, wherein
$\mathbb{Z}_{\scriptsize 2\mbox{gcd}(T_{R},d_{R})}^{\chi}$ is an invertible
part.111Throughout this paper, we use a tilde to distinguish noninvertible
symmetries and operators. Note also that after gauging the vector $U(1)$, the
$\mathbb{Z}_{2}$ part (fermion number) of $\tilde{\mathbb{Z}}_{2T_{R}}^{\chi}$
is part of the gauge symmetry and not a global symmetry. With this in mind, we
continue to denote the noninvertible symmetry by
$\tilde{\mathbb{Z}}_{2T_{R}}^{\chi}$, as this subtlety does not affect our
considerations of the anomaly and spectral degeneracy.  We establish the
noninvertibility through a sequential process starting from the
nonconservation of Noether’s current of the chiral symmetry. Employing the
Hamiltonian formalism and putting the theory on a $3$-dimensional spatial
torus $\mathbb{T}^{3}$, we construct a noninvertible symmetry operator. This
setup provides a simple and explicit route to select the states on which the
operator $\tilde{\mathbb{Z}}_{2T_{R}}^{\chi}$ acts nontrivially.

$SU(N)\times U(1)$ QCD-like theories are naturally endowed with an electric
$1$-form $\mathbb{Z}_{N}^{(1)}$ symmetry acting on the Wilson loops as well as
a $1$-form $U(1)^{(1)}_{m}$ magnetic symmetry that characterizes sectors with
definite magnetic fluxes. Then, it is natural to ask whether the theory
exhibits a ’t Hooft anomaly as we perform a
$\tilde{\mathbb{Z}}_{2T_{R}}^{\chi}$ transformation in the background of
$\mathbb{Z}_{N}^{(1)}$. To address this question, for definiteness222The
conclusions of the work described can be generalized to fermions in higher
representations. we consider a theory with fermions in the $2$-index
symmetric/anti-symmetric representation, which possesses a
$\tilde{\mathbb{Z}}_{2(N\pm 2)}^{\chi}$ noninvertible symmetry. We subject the
theory to $\mathbb{Z}_{N}$ twists (2-form background fields for
$\mathbb{Z}_{N}^{(1)}$) along the non-trivial cycles of $\mathbb{T}^{3}$. In
the presence of the $\mathbb{Z}_{N}$ twists, the noninvertible symmetry
projects onto sectors in Hilbert space with definite magnetic fluxes, easily
identified in the Hamiltonian formalism. For $N$ even, the algebra of
$\tilde{\mathbb{Z}}_{2(N\pm 2)}^{\chi}$ and $\mathbb{Z}_{N}^{(1)}$ fails to
commute by a $\mathbb{Z}_{2}$ phase inside these sectors, revealing a mixed
anomaly between $\mathbb{Z}_{N}^{(1)}$ and $\tilde{\mathbb{Z}}_{2(N\pm
2)}^{\chi}$ symmetries (anomalies involving noninvertible symmetries have
previously been considered in $1+1$ dimensions, see Brunner:2014lua ;
Bhardwaj:2017xup ; Chang:2018iay ; Thorngren:2019iar ; Zhang:2023wlu ;
Choi:2023xjw  and references therein). This anomaly implies that the states in
these special sectors must be $2$-fold degenerate333Similar to the exact
finite-volume degeneracies due to invertible 0-form/1-form anomalies
Cox:2021vsa . on arbitrary size $\mathbb{T}^{3}$. Such degeneracies could be
seen by examining the condensates in realistic lattice simulations.

This paper is organized as follows. In Section 2, we demonstrate the origin of
the noninvertible symmetry by carefully examining $SU(N)\times U(1)$ QCD-like
theories put on $\mathbb{T}^{3}$ with general flux backgrounds and build the
noninvertible operator of $\tilde{\mathbb{Z}}_{2(N\pm 2)}^{\chi}$. Then, we
show that this symmetry has a ’t Hooft anomaly with the $\mathbb{Z}_{N}^{(1)}$
symmetry. In Section 3, we discuss the implication of this anomaly in the
magnetic sectors selected by $\tilde{\mathbb{Z}}_{2(N\pm 2)}^{\chi}$ and
exhibit the exact degeneracy. In the Appendix, we show the equivalence of our
operator construction to the “half-gauging” of a discrete subgroup of the
magnetic one form symmetry $U(1)^{(1)}_{m}$, used in Choi:2022jqy ;
Cordova:2022ieu  to construct a properly normalized defect yielding consistent
Euclidean correlation functions. We conclude with a brief discussion in
Section 4.

##  2 Noninvertible symmetries and their anomalies

Consider $SU(N)\times U(1)$ gauge theory with a single-flavor massless Dirac
fermion in a representation $R$. We use $n$, $T_{R}$, and $d_{R}$ to denote
the $N$-ality, Dynkin index, and the dimension of $R$, respectively, and focus
mostly on $T_{R}=N\pm 2$ for the two-index symmetric (S)/antisymmetric (AS)
representations of $N$-ality $n=2$ and dimension $d_{R}={N(N\pm 1)\over 2}$
for S/AS. Yet, our construction can be easily generalized to theories with
several flavors and fermions in higher representations. Classically, the
theory is endowed with a $U(1)_{\chi}$ global chiral symmetry. The fermion
charges (both are left-handed Weyl) under $(SU(N),U(1),U(1)_{\chi})$ are

$\psi_{R}\sim(R,1,1),~{}~{}\psi_{\bar{R}}\sim(\bar{R},-1,1)\,.$ (1)

The theory with the gauged $U(1)$ has a $\mathbb{Z}_{N}^{(1)}$ $1$-form
electric symmetry, acting on both $SU(N)$ and $U(1)$ Wilson loops, as well as
a $U(1)^{(1)}_{m}$ $1$-form magnetic symmetry which distinguishes the
different $U(1)$-flux sectors.

We use $A$ and $a$ for the $1$-form gauge fields of $SU(N)$ and $U(1)$,
respectively. The corresponding field strengths are $F$ and $f$. The anomaly
equation for the chiral $U(1)_{\chi}$ current is

$\partial_{\mu}j_{\chi}^{\mu}-2T_{R}\partial_{\mu}K^{\mu}_{SU(N)}-{2d_{R}\over
8\pi^{2}}\epsilon_{\mu\nu\lambda\sigma}\partial^{\mu}a^{\nu}\partial^{\lambda}a^{\sigma}=0,$
(2)

where $K^{\mu}_{SU(N)}$ is the $SU(N)$ topological current. Its normalization
is such that the integral of $K^{0}_{SU(N)}\equiv K^{CS}$ over a three-
dimensional manifold changes by an integer under large gauge transformations.
In other words, the operator

$e^{-i2\pi\int_{\mathbb{T}^{3}}d^{3}x\;K^{CS}(A)}$ (3)

is invariant under large gauge transformations. This operator (which shifts
the $\theta$-angle by $2\pi$) will be important in what follows.

When the theory is defined on a spatial manifold with nontrivial $2$-cycles,
e.g., on $\mathbb{T}^{3}$, gauging $U(1)$ breaks $\mathbb{Z}_{2T_{R}}^{\chi}$
of the $SU(N)$ theory further down to $\mathbb{Z}^{\chi}_{\scriptsize
2\mbox{gcd}(d_{R},T_{R})}$. The easiest way to see that is by compactifying
the time direction, so we consider the Euclidean version of the theory on
$\mathbb{T}^{4}$. Under a chiral transformation,
$(\psi_{R},\psi_{\bar{R}})\rightarrow e^{i\alpha}(\psi_{R},\psi_{\bar{R}})$,
the measure changes by

$\displaystyle\exp\left[i\alpha\left(2T_{R}c_{2}(F)+2d_{R}c_{2}(f)\right)\right]\,,$
(4)

where $c_{2}(F)\equiv\int_{\mathbb{T}^{4}}\frac{\,{\rm tr}\,F\wedge
F}{8\pi^{2}}\in\mathbb{Z}$ and
$c_{2}(f)\equiv\int_{\mathbb{T}^{4}}\frac{f\wedge f}{8\pi^{2}}\in\mathbb{Z}$
are the second Chern classes of $SU(N)$ and $U(1)$. Then, demanding that the
phase (4) is trivialized, and using Bézout’s identity, which states that
integers of the form $az_{1}+bz_{2}$ are exact multiples of $\mbox{gcd}(a,b)$,
we arrive at our conclusion.444As a side note, there is no solution of self-
dual BPST instanton on $\mathbb{T}^{4}$ with unit topological charge and zero
twists Braam:1988qk . Adding a twist removes the obstruction to the existence
of the solution. These twists are discussed below. For example, when
gcd($d_{R},T_{R})=1$, there is no residual chiral symmetry and the only
symmetry left over is the $\mathbb{Z}_{2}$ fermion number symmetry which stays
intact, assuming the Lorentz symmetry is unbroken.

It is important to emphasize that $\mathbb{Z}^{\chi}_{\scriptsize
2\mbox{gcd}(d_{R},T_{R})}$ is a genuine invertible symmetry, which is
represented by a unitary operator acting on the Hilbert space of the theory.
What is the fate of $\mathbb{Z}_{2T_{R}}^{\chi}$ that is broken because of the
$U(1)$ instantons? Below, following the approach of Karasik:2022kkq ;
GarciaEtxebarria:2022jky , we argue that this symmetry becomes noninvertible.
We shall exhibit this and study the consequences using Hamiltonian
quantization on $\mathbb{T}^{3}$ in a very explicit manner.

Consider the Hamiltonian quantization of the theory on a rectangular
$\mathbb{T}^{3}$ of sides $L_{1},L_{2},L_{3}$, in the $A_{0}=a_{0}=0$ gauge.
The Hilbert space is constructed in terms of gauge fields $A=A_{i}dx^{i}$ and
$a=a_{i}dx^{i}$, where $i=1,2,3$ is the spatial index (below, we use
$\hat{e}_{i}$ to denote a unit vector in the $i$-th direction). We often use
$x,y,z$ for $x_{1},x_{2},x_{3}$ when writing the components explicitly.

The physical Hilbert space is obtained from field-operator eigenstates after
appropriate gauge averaging and imposing Gauss’s law; a detailed description
can be found in Cox:2021vsa , see also the earlier works
GonzalezArroyo:1987ycm ; vanBaal:2000zc . The gauge fields obey boundary
conditions on $\mathbb{T}^{3}$

$\displaystyle A(x+\hat{e}_{i}L_{i})$ $\displaystyle=$
$\displaystyle\Gamma_{i}A(x)\Gamma_{i}^{-1}\,,$ $\displaystyle
a(x+\hat{e}_{i}L_{i})$ $\displaystyle=$ $\displaystyle
a(x)-d\omega_{i}(x)~{},$ (5)

given in terms of $SU(N)$ and $U(1)$ transition functions, $\Gamma_{i}$ and
$\omega_{i}$, respectively tHooft:1979rtg . We work in a gauge where the
$SU(N)$ transition functions $\Gamma_{i}$ are constant unitary $N\times N$
matrices Gonzalez-Arroyo:1997ugn . The fermions obey similar boundary
conditions (for brevity, we write these for the $N$-ality two case, $n=2$):

$\displaystyle\psi_{R}(x+\hat{e}_{i}L_{i})$ $\displaystyle=$
$\displaystyle\Omega_{i}(x)\Gamma_{i}\;\psi_{R}(x)\;\Gamma_{i}^{T}\,,$
$\displaystyle\psi_{\bar{R}}(x+\hat{e}_{i}L_{i})$ $\displaystyle=$
$\displaystyle\Omega_{i}(x)^{-1}\Gamma_{i}^{*}\;\psi_{\bar{R}}(x)\;\Gamma_{i}^{\dagger}.$
(6)

The $U(1)$ and $SU(N)$ transition functions obey cocycle conditions assuring
that the fields satisfying (5, 6) are single-valued on the torus. The cocycle
conditions obeyed by the transition functions are

$\displaystyle\Gamma_{i}\Gamma_{j}$ $\displaystyle=$ $\displaystyle
e^{i{2\pi\over
N}n_{ij}}\;\Gamma_{j}\Gamma_{i}~{},~{}n_{ij}=-n_{ji},~{}n_{ij}\in\mathbb{Z}\;(\text{mod}N),$
$\displaystyle\Omega_{i}(x+\hat{e}_{j}L_{j})\Omega_{j}(x)$ $\displaystyle=$
$\displaystyle e^{-i{2\pi n\over
N}n_{ij}}\;\Omega_{j}(x+\hat{e}_{i}L_{i})\Omega_{i}(x)\,,$ (7)

or

$\displaystyle\Omega_{i}(x)=e^{i\omega_{i}(x)},\;\text{with}~{}\omega_{i}=\sum\limits_{j=1}^{3}\pi(m_{ij}+{n\over
N}n_{ij}){x_{j}\over L_{j}},~{}m_{ij}=-m_{ji}\in\mathbb{Z}\,.$ (8)

Here, the (mod $N$) integers $n_{ij}$ represent topological classes of
$2$-form $\mathbb{Z}_{N}^{(1)}$ background fields in the respective $2$-planes
and $m_{ij}$ label integer $U(1)$-flux sectors, distinguished by their
magnetic flux through the various 2-planes. The integer $m_{ij}$ are charges
under the global magnetic $U(1)_{m}^{(1)}$ 1-form symmetry Gaiotto:2014kfa .
It is easily seen that all gauge and matter fields are single-valued on
$\mathbb{T}^{3}$ when (7) are obeyed. The integers $n_{ij}$ and $m_{ij}$ label
different flux sectors of the torus Hilbert space.

We next consider the global $\mathbb{Z}_{N}^{(1)}$ symmetry Gaiotto:2014kfa .
The generators of the $1$-form $\mathbb{Z}_{N}^{(1)}$ symmetry act on the
transition functions. We label by $t_{j}$ and $T_{j}$ the $U(1)$ and $SU(N)$
group elements representing the action of the generators of
$\mathbb{Z}_{N}^{(1)}$ in the $j$-th direction on the $U(1)$ and $SU(N)$
transition functions, respectively. The action of $\mathbb{Z}_{N}^{(1)}$ is
given by

$\displaystyle\Omega_{j}$ $\displaystyle\rightarrow$ $\displaystyle
t_{i}(x+\hat{e}_{j}L_{j})\;\Omega_{j}(x)\;t_{i}^{-1}(x)=e^{-i{2\pi n\over
N}\delta_{ij}}\Omega_{j}(x),$ $\displaystyle\Gamma_{j}$
$\displaystyle\rightarrow$ $\displaystyle
T_{i}(x+\hat{e}_{j}L_{j})\;\Gamma_{j}\;T_{i}^{-1}(x)=e^{i{2\pi\over
N}\delta_{ij}}\Gamma_{j}.$ (9)

Put differently, $\mathbb{Z}_{N}^{(1)}$ transformations are represented by
“improper” gauge transformations on the Hilbert space (in the first line,
since both $t_{j}$ and $\Omega_{i}$ are abelian, the $\Omega_{j}$’s can be
dropped).

The explicit expressions for $U(1)$ and $SU(N)$ group elements $t_{j}(x)$ and
$T_{j}(x)$ obeying (2) and generating the center symmetry can be worked out.
The explicit form of $T_{j}(x)$ can be found in the literature555See
GarciaPerez:1992fj ; Poppitz:2022rxv  for $SU(2)$ and Selivanov:2000kg  for
$SU(N)$.. The explicit form of $T_{j}(x)$ depends on both the choice of gauge
for the transition functions and on the $n_{ij}$ 2-form background (this is
because $T_{j}(x)$ have nontrivial winding numbers for $n_{ij}\neq 0$ (mod$N$)
tHooft:1981sps ). We will not need the expression for $T_{j}$, but only the
commutation relation of the $SU(N)$ $\mathbb{Z}_{N}^{(1)}$ center symmetry
generators and the gauge invariant operator (3),

$T_{i}\;e^{-i2\pi\int_{T^{3}}d^{3}xK^{CS}(A)}\;T_{i}^{-1}=e^{-i{2\pi{\epsilon_{ijk}n_{jk}\over
2N}}}\;e^{-i2\pi i\int_{T^{3}}d^{3}K^{CS}(A)}\,,$ (10)

which is nontrivial in the presence of a ’t Hooft twist; see Cox:2021vsa  for
a derivation. On the other hand, the center symmetry generators’ action on the
$U(1)$ can be taken

$\displaystyle t_{i}(x)=e^{-i{2\pi n\over N}{x_{i}\over
L_{i}}}=e^{i\tilde{\lambda}^{(i)}(x)}~{},~{}\text{with}~{}\tilde{\lambda}^{(i)}(x+\hat{e}_{j}L_{j})=\tilde{\lambda}^{(i)}(x)-\delta^{ij}{2\pi
n\over N}~{}.$ (11)

Next, we construct the generator of the noninvertible chiral symmetry. To this
end, we integrate the anomaly equation (2) and use it to define a conserved
(but not gauge-invariant) $U(1)_{\chi}$ symmetry operator on $\mathbb{T}^{3}$.
Because of the boundary conditions in the space directions for the $U(1)$
fields, whose transition functions (8) necessarily depend on $x_{i}$, there
are $a$-dependent boundary terms.666On the other hand, since the transition
functions $\Gamma_{i}$ are constant, no such boundary terms appear for
$SU(N)$. We find, denoting $K^{0}(a)\equiv{1\over
8\pi^{2}}\epsilon^{ijk}a_{i}\partial_{j}a_{k}$:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}^{3}}d^{3}x\left[j_{\chi}^{0}-2T_{R}K^{CS}(A)-2d_{R}K^{0}(a)\right]\bigg{|}^{x_{0}=L_{0}}_{x_{0}=0}-{2d_{R}\over
8\pi^{2}}\int\limits_{0}^{L_{0}}dx_{0}\int
d^{2}S^{i}\epsilon^{ijk}a_{j}\partial_{0}a_{k}\bigg{|}^{x_{i}=L_{i}}_{x_{i}=0}.$

Next, we note that $\partial_{0}a_{k}$ is periodic on $\mathbb{T}^{3}$, while
$a_{k}$ itself obeys (5) with transition function $\omega_{i}$ of (8). It is
easy to see that the second term above is also a total time derivative.
Integrating (2), we finally obtain that
$Q_{\chi}(x_{0}=L_{0})=Q_{\chi}(x_{0}=0)$, where

$\displaystyle Q_{\chi}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}^{3}}d^{3}x\left[j_{\chi}^{0}-2T_{R}K^{CS}(A)-2d_{R}K^{0}(a)\right]$
(13) $\displaystyle+{d_{R}\over 4\pi}(m_{xy}+{n\over
N}n_{xy})\left[\int\limits_{0}^{L_{y}}{dy\over
L_{y}}\int\limits_{0}^{L_{z}}dza_{z}(x=0,y,z)+\int\limits_{0}^{L_{x}}{dx\over
L_{x}}\int\limits_{0}^{L_{z}}dza_{z}(x,y=0,z)\right]$
$\displaystyle+\sum\limits_{\scriptsize\mbox{cyclic}}(x\rightarrow
y\rightarrow z\rightarrow x)\,.$

The last line above indicates that there are two more terms obtained from the
term on the second line by cyclic rotation of $x,y,z$. Again, this is the
operator $Q_{\chi}$ in the sector of Hilbert space with $U(1)$ fluxes $m_{ij}$
and $\mathbb{Z}_{N}^{(1)}$ fluxes $n_{ij}$.

Exponentiating (13), we find the (non-gauge-invariant) operator representing
$\mathbb{Z}_{2T_{R}}^{\chi}$:

$\displaystyle X_{2T_{R}}=e^{i{2\pi\over 2T_{R}}Q_{\chi}}~{}.$ (14)

Let us now study the gauge transformation properties of $X_{2T_{R}}$. First,
we note that because of the gauge invariance of (3), the invariance of (14)
under $SU(N)$ (large and small) gauge transformations is manifest. Next,
consider $U(1)$ gauge transformations with periodic777For use below, we also
recall from (11) that the generator $t_{j}$ of the global
$\mathbb{Z}_{N}^{(1)}$ in the $j$-th direction acts on the $U(1)$ gauge field
as a nonperiodic gauge transformation, i.e. is obtained from (15) upon
replacing $n_{j}\rightarrow-{n\over N}$, where, we remind the reader, $n$ is
the $N$-ality of the matter representation. $e^{i\lambda}$:

$\displaystyle a_{i}\rightarrow
a_{i}-\partial_{i}\lambda,~{}\text{with}~{}\lambda(x+\hat{e}_{i}L_{i})=\lambda(x)+2\pi
n_{i}~{}.$ (15)

To study the transformation properties of (14) under $U(1)$ transformations
(15), we note that, with $\lambda$ from (15),
$-2\pi\int_{T^{3}}d^{3}xK^{0}(a)$ transforms as

$-{1\over
4\pi}\int_{\mathbb{T}^{3}}d^{3}x\epsilon^{ijk}a_{i}\partial_{j}a_{k}\bigg{|}^{a-d\lambda}_{a}={n_{x}\over
2}\int
dydz(\partial_{y}a_{z}-\partial_{z}a_{y})+\sum\limits_{\scriptsize\mbox{cyclic}}(x\rightarrow
y\rightarrow z\rightarrow x)~{}.$ (16)

Since, recalling (5, 7),

$\int dydz(\partial_{y}a_{z}-\partial_{z}a_{y})=-2\pi(m_{yz}+{nn_{yz}\over
N}),$ (17)

we find

$\displaystyle-2\pi\int_{\mathbb{T}^{3}}d^{3}xK^{0}(a)\bigg{|}^{a-d\lambda}_{a}$
$\displaystyle=$ $\displaystyle-{1\over
4\pi}\int_{T^{3}}d^{3}x\epsilon^{ijk}a_{i}\partial_{j}a_{k}\bigg{|}^{a-d\lambda}_{a}=-\pi
n_{x}(m_{yz}+{nn_{yz}\over N})$ (18)
$\displaystyle+\sum\limits_{\scriptsize\mbox{cyclic}}(x\rightarrow
y\rightarrow z\rightarrow x)~{}.$

Likewise, we also find the $U(1)$ gauge transformation of the boundary terms
in (13)

$\int\limits_{0}^{L_{y}}{dy\over
L_{y}}\int\limits_{0}^{L_{z}}dza_{z}(x=0,y,z)+\int\limits_{0}^{L_{x}}{dx\over
L_{x}}\int\limits_{0}^{L_{z}}dza_{z}(x,y=0,z)\bigg{|}^{a-d\lambda}_{a}=-4\pi
n_{z}~{},$ (19)

along with two identical relations obtained by cyclic permutations of $x,y,z$.

Thus, combining the $U(1)$ gauge transformations (18, 19), with the expression
for $Q_{\chi}$ from (13), we find the transformation of $X_{2T_{R}}$

$\displaystyle X_{2T_{R}}[a-d\lambda]=X_{2T_{R}}[a]\;e^{-i2\pi{d_{R}\over
T_{R}}\left[n_{x}(m_{yz}+{nn_{yz}\over N})+n_{y}(m_{zx}+{nn_{zx}\over
N})+n_{z}(m_{xy}+{nn_{xy}\over N})\right]}\,,$ (20)

showing explicitly that the operator (14) is not $U(1)$ gauge invariant.

Thus, following Karasik:2022kkq ; GarciaEtxebarria:2022jky , to make
$X_{2T_{R}}$ gauge invariant, we sum over $n_{x},n_{y},n_{z}$, obtaining the
noninvertible $\tilde{\mathbb{Z}}_{2T_{R}}^{\chi}$ operator

$\displaystyle\tilde{X}_{2T_{R}}=e^{i{2\pi\over
2T_{R}}Q_{\chi}}\sum\limits_{n_{x},n_{y},n_{z}\in Z}e^{-i2\pi{d_{R}\over
T_{R}}\left[n_{x}(m_{yz}+{nn_{yz}\over N})+n_{y}(m_{zx}+{nn_{zx}\over
N})+n_{z}(m_{xy}+{nn_{xy}\over N})\right]}\,,$ (21)

with $Q_{\chi}$ from (13). This equation shows that the operator is
noninvertible and determines the sectors not annihilated by
$\tilde{X}_{2T_{R}}$. To see this, we use the Poisson resummation formula

$\displaystyle\sum_{n_{x}\in Z}e^{-i2\pi{d_{R}\over
T_{R}}(m_{yz}+{nn_{yz}\over N})n_{x}}=\sum_{l_{x}\in Z}\delta\left({d_{R}\over
T_{R}}(m_{yz}+{nn_{yz}\over N})-l_{x}\right)\,.$ (22)

For $\tilde{X}_{2T_{R}}$ to act nontrivially, i.e., not be set to zero, it
must be that in each two-plane, the fluxes $m_{ij},n_{ij}$, $i<j$, have to
obey

${d_{R}\over T_{R}}(m_{yz}+{nn_{yz}\over
N})=l_{x},~{}l_{x}\in\mathbb{Z}\;\;(\text{plus cyclic})\,.$ (23)

It is easy to see that such integer-valued combinations of fluxes always exist
(we shall see examples below). In the Appendix, we construct the operator
$\tilde{X}_{2T_{R}}$ using the “half-gauging” procedure of refs. Choi:2022jqy
; Cordova:2022ieu .

To summarize, here we have constructed a symmetry operator
$\tilde{X}_{2T_{R}}$ of the noninvertible chiral symmetry. The operator of the
noninvertible symmetry acts as a projection operator: it annihilates sectors
of the torus Hilbert space whose fluxes do not obey (23) and acts as unitary
operator in each flux sector obeying (23).888In backgrounds with
$m_{ij},n_{ij}$ chosen to yield integer $l_{i}$ (23), the operator can simply
be defined by (14), since, as (20) shows, for such values of $l_{i}$, it is
gauge invariant.

We next study the commutator of the $\mathbb{Z}_{N}^{(1)}$ center symmetry
transformation with the noninvertible $\tilde{X}_{2T_{R}}$. We denote the
$\mathbb{Z}_{N}^{(1)}$ generator in the $x$-direction by $T_{x}t_{x}$ (for
brevity omitting both hats over operators and the tensor product sign, with
$T_{x}$, $t_{x}$ obeying (2)). We use the commutation relation of (10) of
$T_{i}$ with the operator (3), as well the $U(1)$ transformation law derived
above, eqn. (20), with only $n_{x}$ nonzero and with the integer $n_{x}$
replaced by $-{n\over N}$ (as we reminded the reader in footnote 7). We find

$\displaystyle T_{x}t_{x}\;\tilde{X}_{2T_{R}}\;(T_{x}t_{x})^{-1}$
$\displaystyle=$ $\displaystyle\tilde{X}_{2T_{R}}e^{-i2\pi\left[{n_{yz}\over
N}-{d_{R}\over T_{R}}{n\over N}(m_{yz}+{n{n_{yz}\over N})}\right]}$
$\displaystyle=$ $\displaystyle\tilde{X}_{2T_{R}}e^{-i2\pi\left[{n_{yz}\over
N}-{n\over
N}l_{x}\right]}~{},~{}\text{with}\;l_{x}\;\text{from}\;(\ref{condition1}).$

Equation (2) and its two cyclically permuted versions, constitute our main
result. It shows that—provided both the phase on the r.h.s. is nontrivial and
$\tilde{X}_{2T_{R}}$ is nonzero (i.e. (23) holds)—there is a mixed anomaly
between the $\mathbb{Z}_{N}^{(1)}$ center symmetry and the noninvertible
$\tilde{\mathbb{Z}}_{2T_{R}}^{\chi}$ chiral symmetry of the $SU(N)\times U(1)$
theory. As both operators act nontrivially in the chosen integer-$l_{i}$
$\mathbb{T}^{3}$ Hilbert space, the algebra (2) will be seen to imply exact
degeneracies for any size $\mathbb{T}^{3}$.

In Section 3, we shall show that there are cases where both the phase in (2)
and $\tilde{X}_{2T_{R}}$ are nontrivial, i.e., (23) holds in the appropriate
Hilbert space. As already familiar from Cox:2021vsa , this will be seen to
imply degeneracies in the Hilbert space between different “electric flux”
states (eigenstates of $\mathbb{Z}_{N}^{(1)}$, i.e., of $T_{x}t_{x}$).

Before we continue with analyzing the implications of (2) for the finite-
volume spectrum, let us make a connection of (2) with the Euclidean path
integral. We denote the $\mathbb{T}^{3}$ Hilbert space by ${\cal H}_{l_{i}}$,
with the understanding that $U(1)$ ($m_{ij}$) and $\mathbb{Z}_{N}^{(1)}$
($n_{ij}$) fluxes are chosen to yield an integer-$l_{i}$, so that
$\tilde{X}_{2T_{R}}$ acts nontrivially. We now define the $T_{x}t_{x}$-twisted
partition function via the Hamiltonian formalism as a trace999The relations we
derive below hold also if we insert $(-1)^{F}$ in the partition function. over
states in ${\cal H}_{l_{i}}$ with a $\mathbb{Z}_{N}^{(1)}$ $x$-direction
generator inserted in the partition function, i.e. ${\cal Z}\equiv\,{\rm
tr}\,_{{\cal H}_{l_{i}}}e^{-\beta H}T_{x}t_{x}$. Then, we use the fact that
$\tilde{X}_{2T_{R}}$ acts as an invertible unitary operator in ${\cal
H}_{l_{i}}$, as well as the commutation relation (2), to obtain

$\displaystyle{\cal Z}$ $\displaystyle=$ $\displaystyle\,{\rm tr}\,_{{\cal
H}_{l_{i}}}\left[e^{-\beta H}T_{x}t_{x}\right]=\,{\rm tr}\,_{{\cal
H}_{l_{i}}}\left[e^{-\beta
H}T_{x}t_{x}\tilde{X}_{2T_{R}}\tilde{X}^{\dagger}_{2T_{R}}\right]$ (25)
$\displaystyle=$ $\displaystyle e^{-i2\pi\left[{n_{yz}\over N}-{n\over
N}l_{x}\right]}\;\,{\rm tr}\,_{{\cal H}_{l_{i}}}\left[e^{-\beta
H}T_{x}t_{x}\right]\,.$

We conclude that the chiral symmetry $\tilde{X}_{2T_{R}}$ implies that ${\cal
Z}=e^{-i2\pi\left[{n_{yz}\over N}-{n\over N}l_{x}\right]}{\cal Z}$, so that,
if the phase is nontrivial, ${\cal Z}$ vanishes, unless fermion fields
$(\psi_{\bar{R}}\psi_{R})^{k}$ are inserted to make ${\cal Z}$
nonzero.101010Below, we show that the phase in (25) can take values at most in
$\mathbb{Z}_{2}$, and that $k={N\pm 2\over 2}$, for S/AS fermions, for the
values of $N$ and choice of fluxes where the phase is nontrivial. To obtain a
path integral interpretation of (25), we note that the twisted partition
function (25) sums over $SU(N)$ and $U(1)$ gauge fields which obey the
boundary conditions (5) in the $\mathbb{T}^{3}$ spatial directions, determined
by $n_{ij}$ and $m_{ij}$. The boundary conditions in the Euclidean time
direction (of extent $\beta$) are twisted, by the insertion of $T_{x}$,
leading to a nontrivial $SU(N)$ twist $n_{x4}=1$ in the $x$-time plane. Thus,
the twisted partition function (25) sums over $SU(N)$ field configurations
with topological charges $-{n_{14}n_{yz}\over N}+k=-{n_{yz}\over N}+k$, with
all possible integer $k$ vanBaal:1982ag . On the other hand, the insertion of
$t_{x}$ implies that the $U(1)$ background obeys
$a(x+\hat{e}_{4}\beta)=a(x)+{2\pi n\over N}{dx\over L_{1}}$ and thus
$f_{14}=-{2\pi n\over NL_{1}L_{4}}$. Recalling that the $U(1)$ field strength
in the $yz$ plane is ${2\pi\over L_{2}L_{3}}(m_{yz}+{n\over N}n_{yz})$), the
$U(1)$ topological charge is then equal to ${n\over N}(m_{yz}+{n\over
N}n_{yz})$. Applying a chiral transformation with $\alpha={2\pi\over 2T_{R}}$,
and using the measure transformation (4) (with $c_{2}(F)$, $c_{2}(f)$
substituted by the fractional topological charges just mentioned) we obtain
the phase $e^{-i2\pi[{n_{yz}\over N}-{d_{R}\over T_{R}}{n\over
N}(m_{yz}+{n\over N}n_{yz})]}$, which, after using (23), is seen to be the
same as in (25), as expected.

##  3 Hilbert space, magnetic sectors, and the $2$-fold degeneracy

Here, we analyze the consequences of condition (23) and the algebra of (2).
Condition (23) selects sectors in Hilbert space with definite $U(1)$ magnetic
fluxes in the $2$-$3$, $3$-$1$, and $1$-$2$ planes proportional to the
integers $l_{x},l_{y},l_{z}$, respectively. In the following, we always set
$l_{y}=l_{z}=0$ (with $n_{xy}=m_{xy}=n_{zx}=m_{zx}=0$) to reduce complexity
and examine the theory for various values of $l_{x}={d_{R}\over
T_{R}}(m_{yz}+{nn_{yz}\over N})$. We also choose
$n_{yz}\in\\{0,1,2,...,N-1\\}$. Since we are mainly concerned with symmetric
(S)/antisymmetric (AS) fermions, we set the $N$-ality $n=2$.

We start with sectors with a vanishing magnetic flux, i.e., we set $l_{x}=0$,
which translates into $m_{yz}+\frac{2}{N}n_{yz}=0$. When $N$ is odd, the only
solution is the null solution $m_{yz}=n_{yz}=0$. This yields a trivial phase
in the algebra of (2). However, when $N=2M$ is even, there are two solutions.
First, the null one $m_{yz}=n_{yz}=0$, giving a trivial phase in (2). The
second is the new solution (we denote the phase in (2) by
$e^{-i\alpha}=e^{-2\pi i({n_{yz}\over N}-{2\over N}l_{x})}$):

$\displaystyle\mbox{S/AS}:$ $\displaystyle
N=2M,~{}n_{yz}={M},~{}m_{yz}=-1,~{}l_{x}=0,\alpha=\pi.$ (26)

The $\mathbb{Z}_{2}$ phase in (2) implies that some electric flux states in a
sector with a $0$-magnetic flux are $2$-fold degenerate.

Numerical tests reveal that this pattern continues in sectors with nonzero
magnetic flux, $l_{x}\neq 0$. First, when $N$ is odd, all allowed sectors have
a trivial phase in the algebra of (2), indicating no kinematical constraints
in these theories. For $N$ even, $N=2M$, one can use $l_{x}={d_{R}\over
T_{R}}(m_{yz}+{nn_{yz}\over N})$ to see that there exists integers
$m_{yz}\in\mathbb{Z}$ and $n_{yz}\in\\{0,1,2,...,N-1\\}$ that satisfy the
relation

$\displaystyle M\left(2M\pm 1)(Mm_{yz}+n_{yz}\right)=2M(M\pm 1)l_{x}\,,$ (27)

for S/AS fermions, respectively. Sectors with definite $m_{yz}$ and $n_{yz}$
that satisfy (27) will also yield at most a $\mathbb{Z}_{2}$ phase in the
algebra (2), if they additionally satisfy

$\displaystyle M^{2}\left(n_{yz}+(2M\pm 1)m_{yz}\right)\in M^{2}(M\pm
1)(2\mathbb{Z}+1)\,.$ (28)

The $\mathbb{Z}_{2}$ phase implies that the states in these sectors have
double-fold degeneracy.

Let us flesh this out in detail. For definiteness, we consider S/AS fermions,
which yield a nontrivial phase in (2) as well as involve only a minimal
$SU(N)$ ’t Hooft twist, $n_{yz}=1$ (as opposed to the $l_{x}=0$ solutions
(26), which must have $n_{yz}=N/2$ to produce an anomaly). For both S/AS
fermions, these minimal-twist solutions, yielding an integer $l_{x}$ and a
$\mathbb{Z}_{2}$ phase in (2), must have $N=4p+2$. Using again
$e^{-i\alpha}=e^{-2\pi i({n_{yz}\over N}-{2\over N}l_{x})}$, examples of
$m_{yz}$ that give the $\mathbb{Z}_{2}$ phase are:

$\displaystyle\mbox{S}:$ $\displaystyle
N=4p+2,~{}n_{yz}=1,~{}m_{yz}=-2p-1,~{}l_{x}=-p(3+4p),\alpha=2\pi({1\over
2}+2p),$ $\displaystyle\mbox{AS}:$ $\displaystyle
N=4p+2,~{}n_{yz}=1,~{}m_{yz}=-2p-1,~{}l_{x}=-(1+p)(1+4p),\alpha=2\pi({3\over
2}+2p)\,.$

Thus, we now focus on the flux backgrounds (3), enumerate the degenerate
“electric flux” sectors111111These states can be explicitly constructed in the
semiclassical limit of a small $\mathbb{T}^{3}$ Witten:1982df ; vanBaal:2000zc
(or a small $\mathbb{T}^{2}\in\mathbb{T}^{3}$, with the $\mathbb{T}^{2}$
spanning the $y,z$ directions Tanizaki:2022ngt ; Tanizaki:2022plm ) by
focusing on the lowest energy states. We shall not do this here in complete
detail, but see Footnote 13. in the corresponding Hilbert spaces
${\cal{H}}_{l_{x},l_{y}=l_{z}=0}$, and discuss some of their properties. The
transition functions for $SU(N)$ and $U(1)$ obeying the cocycle conditions
with $n_{yz},m_{yz}$ given in (3) are

$\displaystyle\Gamma_{x}$ $\displaystyle=$ $\displaystyle
1,~{}\Gamma_{y}=P,~{}\Gamma_{z}=Q,~{}\text{where}~{}PQ=e^{i{2\pi\over
N}}QP~{},$ $\displaystyle\omega_{x}$ $\displaystyle=$ $\displaystyle
1,~{}\omega_{y}=-{\pi z\over L_{3}}{4p(p+1)\over 2p+1},~{}\omega_{z}={\pi
y\over L_{2}}{4p(p+1)\over 2p+1}~{}.$ (30)

The center symmetry generators for $SU(N)$ and $U(1)$, obeying (2), can be
taken to be121212For $SU(N)$, these are the ones from Witten:1982df . Briefly,
we remind the reader that $T_{x}$ cannot be taken to be constant, since, as
already discussed, a twist of the partition function in the time direction by
$T_{x}$, recall (25), leads to fractional topological charge on the
$\mathbb{T}^{4}$, equal to ${1\over N}+{\rm{integer}}$. This implies that
$T_{x}$ has fractional winding number tHooft:1981sps , as a map from
$\mathbb{T}^{3}$ to the gauge group, with its $N$-th power being a large gauge
transformation. Thus, on physical states $T_{x}$ obeys $T_{x}^{N}=1$. Explicit
expressions for $T_{1}$ can be found in the literature (see GarciaPerez:1992fj
; Poppitz:2022rxv  for $SU(2)$ and Selivanov:2000kg  for $SU(N)$) but are not
needed here.

$\displaystyle T_{x}$ $\displaystyle=$ $\displaystyle
T_{x}(y,z)~{}(\mbox{recall}\,,T_{x}^{N}|{\rm{phys}}\rangle=|{\rm{phys}}\rangle),T_{y}=Q^{-1},T_{z}=P,$
$\displaystyle t_{k}$ $\displaystyle=$ $\displaystyle e^{-i{2\pi\over
2p+1}{x_{k}\over L_{k}}}~{}.$ (31)

Since $T_{x}t_{x}$ is a symmetry, eigenstates of the Hamiltonian can be
labeled by its eigenvalues, $T_{x}t_{x}|E,e_{x}\rangle=|E,e_{x}\rangle
e^{i{2\pi\over N}e_{x}}$, with $e_{x}\in\mathbb{Z}$ (mod$N$). On the other
hand, the algebra (2),
$T_{x}t_{x}\tilde{X}_{2T_{R}}(T_{x}t_{x})^{-1}=-\tilde{X}_{2T_{R}}$, implies
that, in ${\cal{H}}_{l_{x},l_{y}=l_{z}=0}$, with $l_{x}$ from (3),

$\displaystyle\tilde{X}_{2T_{R}}|E,e_{x}\rangle_{{\cal{H}}_{l_{x},l_{y}=l_{z}=0}}\sim|E,e_{x}+{N\over
2}({\rm mod}\,N)\rangle_{{\cal{H}}_{l_{x},l_{y}=l_{z}=0}}.$ (32)

Thus, $\tilde{X}_{2T_{R}}$ maps an eigenstate of the Hamiltonian of energy $E$
and flux $e_{x}$ to another eigenstate of the same energy, but with flux
$e_{x}+{N\over 2}$ (mod$N$) (hence, a $T_{x}t_{x}$ eigenvalue differing by
$e^{i\pi}$, as per (2); we also note that the phase in the action of
$\tilde{X}$ also depends on whether the state is bosonic of
fermionic).131313As promised, on a small $\mathbb{T}^{3}$, the lowest flux
states can be worked out classically. For the bosonic backgrounds, the lowest-
energy gauge field backgrounds are  $\displaystyle A^{(l)}$ $\displaystyle=$
$\displaystyle-iT_{1}^{l}dT_{1}^{-l},~{}l=0,...,N-1,$ $\displaystyle a^{(l)}$
$\displaystyle=$ $\displaystyle{4\pi p(p+1)\over L_{3}L_{2}(2p+1)}(ydz-
zdy)+{2\pi l\over(2p+1)L_{1}}dx\,.$ (33) The fundamental $SU(N)$ winding
Wilson loops then take values $W_{y}=W_{z}=0,W_{x}=e^{i{2\pi\over N}l}$. Using
the backgrounds (13), solving for the fermions, imposing Gauss’s law, and
averaging over gauge transformations, one can construct the $N$ classically-
degenerate states in ${\cal{H}}_{l_{x},l_{y}=l_{z}=0}$. It is already clear
from (13) that these $N$ states are obtained by the action of $T_{x}t_{x}$
from each other. The electric flux states from eqn. (32) are a discrete
Fourier transform thereof. The anomaly implies that the $N$-fold degeneracy
will be lifted and that only the pairwise degeneracy will remain quantum
mechanically. Extending this small-torus explicit analysis further, along the
lines of Tanizaki:2022ngt ; Tanizaki:2022plm , as well as similar studies for
other backgrounds, e.g., (26), are left for the future.

To further characterize the degenerate flux states, we will show that the
degenerate states (32) have nonvanishing expectation values of a condensate,
which we write schematically as $(\psi_{\bar{R}}\psi_{R})^{N\pm 2\over 2}$.
These expectation values take opposite values in the two degenerate flux
states. To this end, we now go back to our twisted partition function (25).
Since in ${\cal{H}}_{l_{x},l_{y}=l_{z}=0}$ the phase is in $\mathbb{Z}_{2}$,
in order to obtain a nonzero phase we must insert
$(\psi_{\bar{R}}\psi_{R})^{N\pm 2\over 2}$, a gauge invariant object which
transforms with a $\mathbb{Z}_{2}$ phase under $\psi_{R,\bar{R}}\rightarrow
e^{i{2\pi\over 2(N\pm 2)}}\psi_{R,\bar{R}}$. We have

$\displaystyle\langle(\psi_{\bar{R}}\psi_{R})^{N\pm 2\over 2}\rangle$
$\displaystyle=$ $\displaystyle\,{\rm tr}\,_{{\cal H}_{l_{i}}}\left[e^{-\beta
H}(\psi_{\bar{R}}\psi_{R})^{N\pm 2\over 2}T_{x}t_{x}\right]$ (34)
$\displaystyle=$ $\displaystyle\sum_{E,e_{1}=0,...,N-1}e^{-\beta
E(e_{1})}e^{i{2\pi\over N}e_{1}}\langle E,e_{1}|(\psi_{\bar{R}}\psi_{R})^{N\pm
2\over 2}|E,e_{1}\rangle\,.$

Next, recall that $\tilde{X}_{2T_{R}}^{\dagger}\tilde{X}_{2T_{R}}=1$ in
${\cal{H}}_{l_{i}}$, remembering that we are in a definite magnetic flux
sector, where the chiral symmetry operator $\tilde{X}_{2T_{R}}$ acts in an
invertible manner, and using (32), we find that gauge-invariant condensates
obey

$\displaystyle\langle
E,e_{1}|(\psi_{\bar{R}}\psi_{R})^{k}|E,e_{1}\rangle=e^{i{2\pi\over N\pm
2}k}\;\langle E,e_{1}+{N\over 2}|(\psi_{\bar{R}}\psi_{R})^{k}|E,e_{1}+{N\over
2}\rangle~{}.$ (35)

In particular, (35) shows that the condensate appearing in (34) takes opposite
values in the degenerate states. In the twisted partition function (34), this
minus sign is cancelled by change of the phase $e^{i{2\pi\over N}e_{1}}$ (from
the action of $T_{x}t_{x}$). Thus, we can restrict the evaluation of (34) by
summing over half the $e_{1}$ sectors:

$\displaystyle\langle(\psi_{\bar{R}}\psi_{R})^{N\pm 2\over 2}\rangle$
$\displaystyle=$ $\displaystyle 2\sum_{E,e_{1}=0,...{N\over 2}-1}e^{-\beta
E(e_{1})}e^{i{2\pi\over N}e_{1}}\langle E,e_{1}|(\psi_{\bar{R}}\psi_{R})^{N\pm
2\over 2}|E,e_{1}\rangle~{}.$ (36)

These nonzero expectation values can be computed semiclassically at a small
torus and shown not to vanish, similar to Anber:2022qsz .

To summarize, above we constructed the doubly-degenerate states using the
background (3), in the $N=4p+2$ theory, as an example. However, based on the
$\mathbb{Z}_{2}$-valued anomaly, similar descriptions involving degenerate
states with opposite values of the relevant condensate hold in all even-$N$
cases. In particular, the doubly-degenerate flux states corresponding to (26)
can also be explicitly worked out.

Before we discuss these cases, let us contrast the findings in the
$SU(N)\times U(1)$ theory on $\mathbb{T}^{3}$ with those in the $SU(N)$
theory, also on $\mathbb{T}^{3}$. Consider the $SU(4p+2)$ theory (i.e., with
even $N$ not divisible by $4$). It has a $\mathbb{Z}_{2}^{(1)}$ center
symmetry and an invertible discrete chiral symmetry $\mathbb{Z}_{2(N\pm
2)}^{\chi}$. These have a $\mathbb{Z}_{2}$-valued mixed anomaly in appropriate
$\mathbb{Z}_{2}^{(1)}$ backgrounds on $\mathbb{T}^{3}$. This anomaly implies,
as in Cox:2021vsa , an exact two-fold degeneracy in the twisted Hilbert space
of the $SU(4p+2)$ theory, on any torus size. This is similar to the degeneracy
of the $SU(4p+2)\times U(1)$ theory on $\mathbb{T}^{3}$ discussed in this
paper. We stress, however, that the latter theory has gcd$(d_{R},T_{R})=1$,
and hence no genuine chiral symmetry. The degeneracy we found is, thus, due to
the noninvertible chiral $\tilde{\mathbb{Z}}_{2T_{R}}^{\chi}$ symmetry.

Consider now the case when $N$ is divisible by $4$. In the $SU(N)$ theory this
mixed anomaly is trivial on $\mathbb{T}^{3}$ and hence one cannot use the
$\mathbb{Z}_{2}^{(1)}$ $1$-form symmetry to argue for an exact degeneracy on a
finite-size torus. In the $SU(N)\times U(1)$ theory, however, for any even
$N$, we showed that there is an anomaly between the $\mathbb{Z}_{N}^{(1)}$
center and noninvertible chiral symmetries in the (26) background, leading to
an exact two-fold degeneracy at any size $\mathbb{T}^{3}$.

The case of minimal dimension condensate occurs if we take $N=4$ and an
antisymmetric tensor. Here, the anomaly predicts equal and opposite values of
the bilinear fermion condensate $\psi_{\bar{R}}\psi_{R}$ in the two states
that are degenerate at any finite volume, in the appropriately twisted
background. Since the degeneracy is present at any finite volume, should the
condensate remain nonzero in the infinite volume limit, this predicts the
$\mathbb{Z}_{4}\rightarrow\mathbb{Z}_{2}$ (noninvertible) chiral symmetry
breaking in the thermodynamic limit.

We stress that the use of appropriate twists—the $m_{ij},n_{ij}$ with integer
$l_{i}$, i.e. the ones that reveal the anomaly—at finite volume is simply a
tool to probe the gauge dynamics. At least in the theory with a nonzero mass
gap, the infinite volume limit is expected to be independent of the boundary
conditions and the degeneracies revealed are expected to persist in the
thermodynamic limit.

##  4 Discussion

Here, we found that the anomaly establishes the $2$-fold degeneracy on
arbitrary-size $\mathbb{T}^{3}$. Yet, one eventually wants to see what happens
as we take the thermodynamic limit by sending the volume of $\mathbb{T}^{3}$
to infinity.

To speculate on what could happen in $SU(N)\times U(1)$ theory, let us again
return to its cousin, the $SU(N)$ gauge theory with S/AS fermions. The latter
has a global $U(1)$ baryon number and $U(1)_{\chi}$ chiral symmetry. As usual,
quantum effects break $U(1)_{\chi}$ down to, now, the invertible
$\mathbb{Z}_{2(N\pm 2)}^{\chi}$. The $0$-form faithful global symmetry of this
theory is $\frac{U(1)}{\mathbb{Z}_{N\over p}}\times\mathbb{Z}_{2(N\pm
2)}^{\chi}$, with $p=\mbox{gcd}(N,2)$. The fact that the quotient group is
nontrivial means that we can activate a ’t Hooft flux in the center of $SU(N)$
accompanied by a flux in $U(1)$ such that the cocycle conditions are always
obeyed on general four-dimensional manifolds. The authors constructed these
fluxes in vector-like theories, dubbed as the baryon-color (BC) fluxes, in
Anber:2019nze ; Anber:2020gig  (also see Anber:2021lzb ; Anber:2020xfk ;
Anber:2020qzb ; Lohitsiri:2022jyz ; Nakajima:2022jxg  for applications, and
Anber:2021iip  for the construction and applications of these fluxes in chiral
gauge theories). The partition function acquires a $\mathbb{Z}_{N\pm 2}$ phase
as we apply a $\mathbb{Z}_{2(N\pm 2)}^{\chi}$ rotation in the background of
the BC flux. This phase was interpreted as an anomaly of $\mathbb{Z}_{2(N\pm
2)}^{\chi}$ in the BC background. Assuming that the theory has a mass gap and
forms hadrons in the IR, the anomaly is interpreted to imply the existence of
$N\pm 2$ degenerate vacua Anber:2021lzb . One expects to see this degeneracy
emerge on a finite-volume manifold (larger than the inverse strong-scale) and
persist in the thermodynamic limit.

As we argued in this work, gauging the $U(1)$ baryon symmetry endows the
theory with a $\mathbb{Z}_{2}$ anomaly phase when $N$ is even, implying that
at any finite volume there is exact $2$-fold degeneracy. Thus, $2$ degenerate
vacua are guaranteed to survive the infinite volume limit. If the $SU(N)\times
U(1)$ theory also has $N\pm 2$ degenerate vacua in the infinite volume limit
(as the $SU(N)$ theory is believed to) the exact $N\pm 2$-fold degeneracy
should be revealed in the thermodynamic limit.

One might, of course, wonder whether a stronger phase (and stronger
constraints, as in Cordova:2019jqi ) can be exhibited if we subject the
$SU(N)\times U(1)$ theory to a gravitational background. In other words, it
would be interesting to investigate whether there is a mixed anomaly between
the noninvertible $\tilde{\mathbb{Z}}_{2(N\pm 2)}^{\chi}$ and gravity. If the
gravitational anomaly does not produce a stronger phase beyond
$\mathbb{Z}_{2}$, the implications in the thermodynamic limit of the
$SU(N)\times U(1)$ theory are of interest. By weakly gauging $U(1)$ in the
$SU(N)$ theory, one anticipates the presence of $N\pm 2$ nearly degenerate
vacua. However, the dynamics of the $U(1)$ gauge field may affect the precise
degeneracy. This intriguing investigation remains open for future study.

Finally, we comment on the $SU(N)\times U(1)$ theory at finite temperature.
The presence of the identified mixed anomaly implies that it is necessary for
either the $1$-form symmetry $\mathbb{Z}_{N}^{(1)}$, the $0$-form symmetry
$\tilde{\mathbb{Z}}_{2(N\pm 2)}^{\chi}$, or both symmetries to be broken
Gaiotto:2017yup ; Shimizu:2017asf ; Komargodski:2017smk . Actually, since
$\mathbb{Z}_{N}^{(1)}$ acts on the $U(1)$ Wilson lines, we expect it to be
broken at zero and finite temperature. Consequently, the anomaly is
consistently matched, leading us to only expect the restoration of the broken
$\tilde{\mathbb{Z}}_{2(N\pm 2)}^{\chi}$ symmetry and the breaking of the
$\mathbb{Z}_{2}^{(1)}$ subgroup of the 1-form symmetry for even-$N$ at some
finite temperature.

Acknowledgments:

M.A. is supported by STFC through grant ST/T000708/1. E.P. is supported by a
Discovery Grant from NSERC. We thank Aleksey Cherman, Theo Jacobson, and Emily
Nardoni for discussions. We also thank an anonymous referee for the suggestion
to consider the “half-gauging” construction of the noninvertible defect and
study its relation to the Hilbert space construction, see Appendix A.

##  Appendix A The noninvertible defect via “half-gauging” of
$\mathbb{Z}_{T_{R}}^{(1)}\subset U(1)^{(1)}_{m}$

Here, we consider the construction of the noninvertible defect for141414Once
again, we acknowledge our abuse of notation: as per the remark of Footnote 1,
the $\mathbb{Z}_{2}$ fermion number subgroup of ${\mathbb{Z}}_{2T_{R}}^{\chi}$
is part of the $U(1)$ gauge group. $\tilde{\mathbb{Z}}_{2T_{R}}^{\chi}$ using
the “half-gauging” procedure of refs. Choi:2022jqy ; Cordova:2022ieu  instead
of the sum over gauge orbits of Karasik:2022kkq ; GarciaEtxebarria:2022jky .
The advantage of the former construction is that it gives rise to well-defined
Euclidean correlation functions involving the noninvertible defect, considered
more generally than as an operator inserted at a particular time. In
particular, this allows inserting the defect at a particular location in
space, giving rise to well-defined Hilbert spaces twisted by the noninvertible
symmetry.

Our discussion below makes use of the techniques described explicitly in ref.
Choi:2022jqy  and is restricted to the case gcd$(d_{R},T_{R})=1$. For our
$SU(N)\times U(1)$ theories with a two-index S/AS Dirac fermion, this is the
case of even-$N$ not divisible by $4$, leaving the generalization to the more
general case for future work (although a generalization to
gcd$(d_{R},T_{R})>1$ should be possible Choi:2022jqy ). In order to define the
defect, we consider the gauging of the ${\mathbb{Z}}_{T_{R}}^{(1)}$ subgroup
of the magnetic $U(1)_{m}^{(1)}$ 1-form symmetry. As shown in Choi:2022jqy ,
this gauging produces the same theory but with a discrete shift of the $U(1)$
theta angle $\theta\rightarrow\theta-{2\pi d_{R}\over T_{R}}$. This shift, as
per our Eqn. (4), is undone by a $\mathbb{Z}_{2T_{R}}$ chiral rotation of the
fermions $(\psi_{R},\psi_{\bar{R}})$. We conclude that the $SU(N)\times U(1)$
theory is invariant under the above gauging of
$\mathbb{Z}_{T_{R}}^{(1)}\subset U(1)_{m}^{(1)}$.

As in Choi:2022jqy ; Cordova:2022ieu , the upshot of the half-gauging
procedure is to define a defect, replacing our eqn. (21) for the operator
$\tilde{X}_{2T_{R}}$ by the following object, which we label, for brevity, by
the same letter

$\displaystyle\tilde{X}_{2T_{R}}$ $\displaystyle=$ $\displaystyle
e^{i{2\pi\over
2T_{R}}\int\limits_{t=0}d^{3}xj_{\chi}^{0}-i2\pi\int\limits_{t=0}d^{3}xK^{CS}(A)+\int\limits_{t=0}{\cal
A}^{T_{R},d_{R}}[{da\over T_{R}}]}~{},$ (37)

where, as in the main text, $a$ is the dynamical $U(1)$ gauge field and $A$ is
the $SU(N)$ gauge field with $K^{CS}[A]$ entering as in (3).

The 3d defect TQFT ${\cal A}^{T_{R},d_{R}}[{da\over T_{R}}]$ is defined via an
integral over a 4d bulk with a boundary, which is here taken to be the $t=0$
plane:

$\displaystyle e^{\;\;\int\limits_{t=0}{\cal A}^{T_{R},d_{R}}[{da\over
T_{R}}]}$ $\displaystyle=$ $\displaystyle\int{\cal
D}(b,c)\;e^{\;\;\int\limits_{t\geq 0}\left({id_{R}\over 4\pi T_{R}}\;da\wedge
da+{i\over 2\pi}\;b^{(2)}\wedge da+{iT_{R}\over 2\pi}\;b^{(2)}\wedge
dc^{(1)}+{ikT_{R}\over 4\pi}\;b^{(2)}\wedge b^{(2)}\right)}.$

The fields $(b^{(2)},c^{(1)})$ define the $2$-form $\mathbb{Z}_{T_{R}}$ gauge
field (used in the half-gauging procedure) and obeying the Dirichlet boundary
condition $b^{(2)}=0$ at $t=0$, and $k$ is the modular inverse of $d_{R}$,
i.e. $kd_{R}=1\;({\rm mod}T_{R})$. Defining the defect via the “half-gauging”
procedure replaces the sum over gauge copies of the gauge noninvariant $U(1)$
Chern-Simons term by the TQFT ${\cal A}^{T_{R},d_{R}}[{da\over T_{R}}]$ and
produces a well defined Euclidean defect.

The ${\cal A}^{T_{R},d_{R}}[{da\over T_{R}}]$ theory has a
${\mathbb{Z}}_{T_{R}}^{(1)}$ global symmetry with an anomaly $d_{R}$, and the
partition function $e^{\;\int\limits_{t=0}{\cal A}^{T_{R},d_{R}}[{da\over
T_{R}}]}$ of (A) has the transformation properties of $e^{-i{d_{R}T_{R}\over
4\pi}\int\limits_{t\geq 0}{da\over T_{R}}\wedge{da\over T_{R}}}$. In
particular, under regular gauge transformations ${\delta a=d\omega}$ with
$\oint d\omega=2\pi{\mathbb{Z}}$,

$\displaystyle\delta\;e^{\;\int\limits_{t=0}{\cal A}^{T_{R},d_{R}}[{da\over
T_{R}}]}=e^{i{d_{R}\over T_{R}}\int\limits_{t=0}{d\omega}\wedge{da\over
2\pi}}\;e^{\;\int\limits_{t=0}{\cal A}^{T_{R},d_{R}}[{da\over T_{R}}]}~{},$
(39)

which implies, together with the gauge invariance of ${\cal
A}^{T_{R},d_{R}}[{da\over T_{R}}]$, that the defect vanishes unless
${d_{R}\over T_{R}}\oint da=2\pi{\mathbb{Z}}$. This is precisely the condition
(23) of the main text, in the absence of an electric ${\mathbb{Z}}_{N}^{(1)}$
center symmetry background. Thus, we have succeeded in defining a
noninvertible defect associated with the $\tilde{\mathbb{Z}}_{2T_{R}}^{\chi}$
chiral symmetry.

The question left open, then, is to show that the definition of
$\tilde{X}_{2T_{R}}$ from (37) reproduces the mixed anomaly with center
symmetry exhibited in Eqn. (2). This would necessitate generalizing the half-
gauging procedure to backgrounds with fractional $\oint
da\in{2\pi{\mathbb{Z}}\over N}$.151515Thus, incorporating the electric
${\mathbb{Z}}_{N}^{(1)}$ center-symmetry background, denoted by $n_{yz}$ in
(23, 2). In the language of defects, one has to consider defects associated
with the (invertible) electric ${\mathbb{Z}}_{N}^{(1)}$ 1-form symmetry of the
$SU(N)\times U(1)$ theory and determine the fusion rules of these codimension-
two defects with the noninvertible defect defined by (37). The fusion rules of
$\tilde{X}_{2T_{R}}$ with these codimension-two
${\mathbb{Z}}_{N}^{(1)}$-defects of two different orientations are relevant to
the anomaly (2). The first are “perpendicular” to $\tilde{X}_{2T_{R}}$, with a
worldvolume in the $x$-$t$ plane and in the language of this paper correspond
to the turning on of $n_{yz}$ from (2). The other type of codimension-two
defects implementing the ${\mathbb{Z}}_{N}^{(1)}$ symmetry are “parallel” to
$\tilde{X}_{2T_{R}}$, with world volume in the $y$-$z$ plane,161616We note
that examples of fusion rules for similar parallel codimension-two symmetry
defects with duality/triality defects were studied in Choi:2022zal .
implementing the one-form symmetry transformation with parameter
$n_{x}=-{n\over N}$ due to $T_{x}t_{x}$ of (2). The careful study of the
various defects mentioned should make finding these fusion rules possible, but
we leave this interesting question to future studies. We believe, however,
that the ${\mathbb{T}}^{4}$ path-integral discussion after Eqn. (25) gives
strong support for our finding (2)—while, admittedly, leaving the study of the
more general interesting situations mentioned above open.

## References

  * (1) D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172, [arXiv:1412.5148].
  * (2) J. Fuchs, I. Runkel, and C. Schweigert, TFT construction of RCFT correlators 1. Partition functions, Nucl. Phys. B 646 (2002) 353–497, [hep-th/0204148].
  * (3) Z. Komargodski, K. Ohmori, K. Roumpedakis, and S. Seifnashri, Symmetries and strings of adjoint QCD2, JHEP 03 (2021) 103, [arXiv:2008.07567].
  * (4) M. Nguyen, Y. Tanizaki, and M. Ünsal, Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond $N$-ality, JHEP 03 (2021) 238, [arXiv:2101.02227].
  * (5) M. Nguyen, Y. Tanizaki, and M. Ünsal, Noninvertible 1-form symmetry and Casimir scaling in 2D Yang-Mills theory, Phys. Rev. D 104 (2021), no. 6 065003, [arXiv:2104.01824].
  * (6) Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam, and S.-H. Shao, Noninvertible duality defects in 3+1 dimensions, Phys. Rev. D 105 (2022), no. 12 125016, [arXiv:2111.01139].
  * (7) J. Wang and Y.-Z. You, Gauge Enhanced Quantum Criticality Between Grand Unifications: Categorical Higher Symmetry Retraction, arXiv:2111.10369.
  * (8) L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki, and A. Tiwari, Non-invertible higher-categorical symmetries, SciPost Phys. 14 (2023), no. 1 007, [arXiv:2204.06564].
  * (9) Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam, and S.-H. Shao, Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions, arXiv:2204.09025.
  * (10) J. Kaidi, G. Zafrir, and Y. Zheng, Non-invertible symmetries of $\mathcal{N}$ = 4 SYM and twisted compactification, JHEP 08 (2022) 053, [arXiv:2205.01104].
  * (11) Y. Choi, H. T. Lam, and S.-H. Shao, Noninvertible Global Symmetries in the Standard Model, Phys. Rev. Lett. 129 (2022), no. 16 161601, [arXiv:2205.05086].
  * (12) C. Cordova and K. Ohmori, Noninvertible Chiral Symmetry and Exponential Hierarchies, Phys. Rev. X 13 (2023), no. 1 011034, [arXiv:2205.06243].
  * (13) Y. Choi, H. T. Lam, and S.-H. Shao, Noninvertible Time-Reversal Symmetry, Phys. Rev. Lett. 130 (2023), no. 13 131602, [arXiv:2208.04331].
  * (14) T. Bartsch, M. Bullimore, A. E. V. Ferrari, and J. Pearson, Non-invertible Symmetries and Higher Representation Theory I, arXiv:2208.05993.
  * (15) J. J. Heckman, M. Hübner, E. Torres, and H. Y. Zhang, The Branes Behind Generalized Symmetry Operators, Fortsch. Phys. 71 (2023), no. 1 2200180, [arXiv:2209.03343].
  * (16) C. Cordova, S. Hong, S. Koren, and K. Ohmori, Neutrino Masses from Generalized Symmetry Breaking, arXiv:2211.07639.
  * (17) A. Karasik, On anomalies and gauging of U(1) non-invertible symmetries in 4d QED, arXiv:2211.05802.
  * (18) I. n. García Etxebarria and N. Iqbal, A Goldstone theorem for continuous non-invertible symmetries, arXiv:2211.09570.
  * (19) Y. Choi, H. T. Lam, and S.-H. Shao, Non-invertible Gauss Law and Axions, arXiv:2212.04499.
  * (20) R. Yokokura, Non-invertible symmetries in axion electrodynamics, arXiv:2212.05001.
  * (21) L. Bhardwaj, S. Schafer-Nameki, and A. Tiwari, Unifying Constructions of Non-Invertible Symmetries, arXiv:2212.06159.
  * (22) L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki, and A. Tiwari, Non-Invertible Symmetry Webs, arXiv:2212.06842.
  * (23) T. Bartsch, M. Bullimore, A. E. V. Ferrari, and J. Pearson, Non-invertible Symmetries and Higher Representation Theory II, arXiv:2212.07393.
  * (24) J. J. Heckman, M. Hubner, E. Torres, X. Yu, and H. Y. Zhang, Top Down Approach to Topological Duality Defects, arXiv:2212.09743.
  * (25) A. Apte, C. Cordova, and H. T. Lam, Obstructions to Gapped Phases from Non-Invertible Symmetries, arXiv:2212.14605.
  * (26) C. Delcamp and A. Tiwari, Higher categorical symmetries and gauging in two-dimensional spin systems, arXiv:2301.01259.
  * (27) J. Kaidi, E. Nardoni, G. Zafrir, and Y. Zheng, Symmetry TFTs and Anomalies of Non-Invertible Symmetries, arXiv:2301.07112.
  * (28) P. Putrov and J. Wang, Categorical Symmetry of the Standard Model from Gravitational Anomaly, arXiv:2302.14862.
  * (29) M. Dierigl, J. J. Heckman, M. Montero, and E. Torres, R7-Branes as Charge Conjugation Operators, arXiv:2305.05689.
  * (30) I. Brunner, N. Carqueville, and D. Plencner, Discrete torsion defects, Commun. Math. Phys. 337 (2015), no. 1 429–453, [arXiv:1404.7497].
  * (31) L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions, JHEP 03 (2018) 189, [arXiv:1704.02330].
  * (32) C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, Topological Defect Lines and Renormalization Group Flows in Two Dimensions, JHEP 01 (2019) 026, [arXiv:1802.04445].
  * (33) R. Thorngren and Y. Wang, Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases, arXiv:1912.02817.
  * (34) C. Zhang and C. Córdova, Anomalies of $(1+1)D$ categorical symmetries, arXiv:2304.01262.
  * (35) Y. Choi, B. C. Rayhaun, Y. Sanghavi, and S.-H. Shao, Comments on Boundaries, Anomalies, and Non-Invertible Symmetries, arXiv:2305.09713.
  * (36) A. A. Cox, E. Poppitz, and F. D. Wandler, The mixed 0-form/1-form anomaly in Hilbert space: pouring the new wine into old bottles, JHEP 10 (2021) 069, [arXiv:2106.11442].
  * (37) P. J. Braam and P. van Baal, Nahm’s Transformation for Instantons, Commun. Math. Phys. 122 (1989) 267.
  * (38) A. González Arroyo and C. P. Korthals Altes, The Spectrum of Yang-Mills Theory in a Small Twisted Box, Nucl. Phys. B 311 (1988) 433–449.
  * (39) P. van Baal, QCD in a finite volume, hep-ph/0008206.
  * (40) G. ’t Hooft, A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories, Nucl. Phys. B 153 (1979) 141–160.
  * (41) A. González-Arroyo, Yang-Mills fields on the four-dimensional torus. Part 1.: Classical theory, in Advanced Summer School on Nonperturbative Quantum Field Physics, pp. 57–91, 6, 1997.  hep-th/9807108.
  * (42) M. García Pérez and A. González-Arroyo, Numerical study of Yang-Mills classical solutions on the twisted torus, J. Phys. A 26 (1993) 2667–2678, [hep-lat/9206016].
  * (43) E. Poppitz and F. D. Wandler, Gauge theory geography: charting a path between semiclassical islands, JHEP 02 (2023) 014, [arXiv:2211.10347].
  * (44) K. G. Selivanov and A. V. Smilga, Classical Yang-Mills vacua on T**3: Explicit constructions, Phys. Rev. D 63 (2001) 125020, [hep-th/0010243].
  * (45) G. ’t Hooft, Aspects of Quark Confinement, Phys. Scripta 24 (1981) 841–846.
  * (46) P. van Baal, Some Results for SU(N) Gauge Fields on the Hypertorus, Commun. Math. Phys. 85 (1982) 529.
  * (47) E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B 202 (1982) 253.
  * (48) Y. Tanizaki and M. Ünsal, Center vortex and confinement in Yang–Mills theory and QCD with anomaly-preserving compactifications, PTEP 2022 (2022), no. 4 04A108, [arXiv:2201.06166].
  * (49) Y. Tanizaki and M. Ünsal, Semiclassics with ’t Hooft flux background for QCD with 2-index quarks, JHEP 08 (2022) 038, [arXiv:2205.11339].
  * (50) M. M. Anber and E. Poppitz, The gaugino condensate from asymmetric four-torus with twists, JHEP 01 (2023) 118, [arXiv:2210.13568].
  * (51) M. M. Anber and E. Poppitz, On the baryon-color-flavor (BCF) anomaly in vector-like theories, JHEP 11 (2019) 063, [arXiv:1909.09027].
  * (52) M. M. Anber and E. Poppitz, Generalized ’t Hooft anomalies on non-spin manifolds, JHEP 04 (2020) 097, [arXiv:2002.02037].
  * (53) M. M. Anber, Condensates and anomaly cascade in vector-like theories, JHEP 03 (2021) 191, [arXiv:2101.04132].
  * (54) M. M. Anber and E. Poppitz, Deconfinement on axion domain walls, JHEP 03 (2020) 124, [arXiv:2001.03631].
  * (55) M. M. Anber and S. Baker, Natural inflation, strong dynamics, and the role of generalized anomalies, Phys. Rev. D 102 (2020), no. 10 103515, [arXiv:2008.05491].
  * (56) N. Lohitsiri and T. Sulejmanpasic, Comments on QCD3 and anomalies with fundamental and adjoint matter, JHEP 10 (2022) 081, [arXiv:2205.07825].
  * (57) T. Nakajima, T. Sakai, and R. Yokokura, BCF anomaly and higher-group structure in the low energy effective theories of mesons, JHEP 01 (2023) 175, [arXiv:2212.12987].
  * (58) M. M. Anber, S. Hong, and M. Son, New anomalies, TQFTs, and confinement in bosonic chiral gauge theories, JHEP 02 (2022) 062, [arXiv:2109.03245].
  * (59) C. Córdova and K. Ohmori, Anomaly Constraints on Gapped Phases with Discrete Chiral Symmetry, Phys. Rev. D 102 (2020), no. 2 025011, [arXiv:1912.13069].
  * (60) D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, Theta, Time Reversal, and Temperature, JHEP 05 (2017) 091, [arXiv:1703.00501].
  * (61) H. Shimizu and K. Yonekura, Anomaly constraints on deconfinement and chiral phase transition, Phys. Rev. D 97 (2018), no. 10 105011, [arXiv:1706.06104].
  * (62) Z. Komargodski, T. Sulejmanpasic, and M. Ünsal, Walls, anomalies, and deconfinement in quantum antiferromagnets, Phys. Rev. B 97 (2018), no. 5 054418, [arXiv:1706.05731].


# A single pulse study of a millisecond pulsar PSR J0621+1002

S. Q. Wang11affiliation: Xinjiang Astronomical Observatory, Chinese Academy of
Sciences, 150 Science 1-Street, Urumqi 830011, People’s Republic of China;
22affiliation: University of Chinese Academy of Sciences, Beijing 100049,
People’s Republic of China 33affiliation: CAS Key Laboratory of FAST, NAOC,
Chinese Academy of Sciences, Beijing 100101, People’s Republic of China , J.
B. Wang11affiliation: Xinjiang Astronomical Observatory, Chinese Academy of
Sciences, 150 Science 1-Street, Urumqi 830011, People’s Republic of China;
44affiliation: Key Laboratory of Radio Astronomy, Chinese Academy of Sciences,
150 Science 1-Street, Urumqi 830011, People’s Republic of China 55affiliation:
Xinjiang Key Laboratory of Radio Astrophysics, 150 Science1-Street, Urumqi,
Xinjiang, 830011, People’s Republic of China , N. Wang11affiliation: Xinjiang
Astronomical Observatory, Chinese Academy of Sciences, 150 Science 1-Street,
Urumqi 830011, People’s Republic of China; 44affiliation: Key Laboratory of
Radio Astronomy, Chinese Academy of Sciences, 150 Science 1-Street, Urumqi
830011, People’s Republic of China 55affiliation: Xinjiang Key Laboratory of
Radio Astrophysics, 150 Science1-Street, Urumqi, Xinjiang, 830011, People’s
Republic of China , Y. Feng66affiliation: National Astronomical Observatories,
Chinese Academy of Sciences, A20 Datun Road, Chaoyang District, Beijing
100101, People’s Republic of China 22affiliation: University of Chinese
Academy of Sciences, Beijing 100049, People’s Republic of China , S. B.
Zhang77affiliation: Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210008, People’s Republic of China , K.J. Lee88affiliation: Kavli
Institute for Astronomy and Astrophysics, Peking University, Beijing 100871,
People’s Republic of China , D. Li66affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District,
Beijing 100101, People’s Republic of China 22affiliation: University of
Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
99affiliation: NAOC-UKZN Computational Astrophysics Centre, University of
KwaZulu-Natal, Durban 4000, South Africa , J. G. Lu66affiliation: National
Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Road,
Chaoyang District, Beijing 100101, People’s Republic of China , J. T.
Xie11affiliation: Xinjiang Astronomical Observatory, Chinese Academy of
Sciences, 150 Science 1-Street, Urumqi 830011, People’s Republic of China;
22affiliation: University of Chinese Academy of Sciences, Beijing 100049,
People’s Republic of China , D. J. Zhou66affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District,
Beijing 100101, People’s Republic of China 22affiliation: University of
Chinese Academy of Sciences, Beijing 100049, People’s Republic of China , L.
Zhang66affiliation: National Astronomical Observatories, Chinese Academy of
Sciences, A20 Datun Road, Chaoyang District, Beijing 100101, People’s Republic
of China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present radio observation of a millisecond pulsar PSR J0621+1002 using the
Five-hundred-meter Aperture Spherical radio Telescope (FAST). The pulsar shows
periodic pulse intensity modulations for both the first and the third pulse
components. The fluctuation spectrum of the first pulse component has one peak
of 3.0$\pm$0.1 pulse periods, while that of the third pulse component has two
diffused peaks of 3.0$\pm$0.1 and 200$\pm$1 pulse periods. The single pulse
timing analysis is carried out for this pulsar and the single pulses can be
divided into two classes based on the post-fit timing residuals. We examined
the achievable timing precision using only the pulses in one class or bright
pulses. However, the timing precision improvement is not achievable.

methods: observational — pulsars: general — pulsars: individual: PSR
J0621+1002

††software: DSPSR (van Straten & Bailes, 2011), PSRCHIVE (Hotan et al., 2004),
TEMPO2 (Hobbs et al., 2006), PSRSALSA (Weltevrede, 2016)

##  1 INDRUCTION

Radio pulsars are known to exhibit various emission properties, such as
nulling (Backer, 1970a; Wang et al., 2007), mode changing (Bartel & Hankins,
1982) and giant pulse (Staelin & Reifenstein, 1968). Some pulsars exhibit
periodic emission variations (Weltevrede, 2016). One well known of such
phenomenon is subpulse drifting, in which the subpulse drifts in pulse
longitude across a pulse sequence (Weltevrede, 2016). Another similar
phenomenon is amplitude modulation, where pulses only show periodic intensity
modulations that do not propagate in pulse phase (Basu et al., 2016).

Periodic amplitude modulation and drifting subpulse are mainly seen in normal
pulsars. These phenomena are difficult to detect in millisecond pulsars (MSPs)
because of their low flux densities (Edwards, & Stappers, 2003). Single-pulse
studies have been only carried out for some bright MSPs (Jenet et al., 1998,
2001; Liu et al., 2015; Wang et al., 2020). Edwards, & Stappers (2003)
reported pulse-to-pulse intensity modulations for six MSPs. Liu et al. (2016)
revealed the diffused subpulse drifting phenomenon of PSR J1713+0747.
Recently, Mahajan et al. (2018) found the first mode changing MSP, PSR
B1957+20.

It is possible to detect nanohertz gravitational waves by monitoring pulse
times of arrival (ToAs) of an ensemble of the most stable MSPs (Shannon et
al., 2013; Arzoumanian et al., 2020). The success of this experiment strongly
depends on the achievable timing precision. On short time scales, pulsar
timing precision is limited by white noise (Liu et al., 2012; Shannon et al.,
2014; Lam et al., 2019). Radiometer noise and phase variation of integrated
profiles induced by pulse-to-pulse variability, commonly referred to as jitter
noise, are the main sources of white noise. Thus, single-pulse studies of MSPs
can provide us a fundamental limit on the achievable timing precision on short
timescales. Highly sensitive radio telescopes, such as the Five hundred meter
Aperture Spherical Telescope (FAST), provide us a great opportunity to study
pulse-to-pulse variability of MSPs and jitter noise.

PSR J0621+1002 is a 28.9 ms MSP monitored by the European Pulsar Timing Array
(EPTA, see Desvignes et al. 2016). The pulsar is in a binary system with an
8.3-days orbital period and a 0.41 $\rm M_{\odot}$ CO white dwarf companion
(Camilo et al., 1996; Kramer et al., 1998; Splaver et al., 2002; Kasian,
2012). In this paper, we present analysis on periodic pulse intensity
modulation and jitter noise of the pulsar with the FAST. In Section 2, we
describe our observation. The results are presented in section 3. We discuss
and summarise our results in Section 4.

##  2 OBSERVATIONS AND DATA PROCESSING

We obtained 54066 single pulses in a 26 minutes observation on January 14 2020
(project ID 3062). The observation was conducted using the central beam of the
19-beam receiver with a frequency range between 1050 and 1450 MHz (Jiang et
al., 2019). The data were recorded in search mode PSRFITS format with 4
polarizations, 8-bit samples of an 49.152-$\mu$s interval, and 4096 frequency
channels.

Individual pulses were extracted using dspsr software package (van Straten &
Bailes 2011). The $-$K option in dspsr was used to remove inter-channel
dispersion delays. 5% of the band-edges and flag narrow-band and impulsive
radio-frequency interference (RFI) are removed using psrchive software package
(Hotan et al. 2004). More details of the RFI environment of FAST are provided
in Jiang et al. (2019). Polarization calibration was achieved by correcting
for the differential gain and phase between the receptors through separate
measurements using a noise diode signal. Flux density was calibrated using
observations of 3C 286 (Baars et al., 1977). Flux density and rotation measure
(RM) are obtained using the psrchive program psrflux and rmfit.

Noise-free standard templates were formed by interactively fitting scaled von
Mises functions (using the psrchive program paas) to a high S/N observed
profile. ToAs were obtained by cross-correlating pulse profile with the
template using the pat command. Timing residuals were calculated using the
tempo2 software package (Hobbs et al., 2006). PSRSALSA package was used to
carry out fluctuation analysis (Weltevrede, 2016).

##  3 RESULTS

###  3.1 The periodical pulse intensity modulation

Figure 1: Polarization profile for PSR J0621+1002. The black, red, and blue
lines are the total intensity, linear polarized intensity, and circular
polarized intensity, respectively. Position angles (black dots) and
corresponding uncertainties (red bars) of the linear polarized emission are
shown as a function of pulse phase.  Figure 2: Single-pulse stack of 300
pulses of PSR J0621+1002. Average pulse profile is shown in upper panel and
divided into three components (labeled as I, II and III) by the four vertical
lines.

Figure 3: Fluctuation analysis of the emission of PSR J0621+1002. Top row: the
LRFS and a side panel showing the horizontally integrated power for the first
(left panel) and third (right panel) pulse components.Bottom row: the 2DFS and
a side panel showing the horizontally integrated power for the first (left
panel) and third (right panel) pulse components.

The polarization profile for PSR J0621+1002 is shown in Figure 1. The overall
pulse width is about 175 degrees, nearly half of the pulse period. The profile
features are consistent with the EPTA observation at 1.4 GHz (Desvignes et
al., 2016). The pulse profile has three components and the fractional linear
polarization for the second component is higher than the first and third
components.

The position angles (PAs) vary significantly across the first and the third
components and it is relatively smooth across the second component. Generally,
for pulse profile with three components, the first and third pulse components
are related to the cone emissions and the second pulse component is associated
with the core emission (Backer, 1976; Rankin, 1983) and the gradient of PA
swing is large near the pulse centre (Lyne & Manchester, 1988). The second
component of PSR J0621+1002 exhibits a relatively smooth PA swing which does
not agree with the core emission. This component also shows high fractional
linear polarization which is characterized by the cone emission. Therefore, we
suggested that all the three components of PSR J0621+1002 are related to the
cone emissions. Our measured flux density at 1250 MHz and RM are 1.90 mJy and
$53.0\pm 0.1\,{\rm rad\,m^{-2}}$, respectively, which are consistent with the
previously published results (Sobey et al., 2019).

Single-pulse stack of 300 pulses is shown in Figure 2. To investigate the
pulse modulation behavior, longitude-resolved fluctuation spectrum (LRFS,
Backer 1970b) and two-dimensional fluctuation spectrum (2DFS, Edwards &
Stappers 2002) was carried out for each pulse component. 2DFS is a useful tool
to determine whether subpulses are drifting in pulse longitude (Weltevrede,
2016) and LRFS can be used to detect periodicity of subpulse modulation. Using
PSRSALSA package with FFT-size of 1024, the LRFS and 2DFS of all the data are
shown in the upper and bottom panels of Figure 3, respectively. There is a
clear periodic pulse intensity modulation for both the first and the third
pulse components. The fluctuation spectrum of the first pulse component has a
clear peak of $3.0\pm 0.1\,P$ with the pulse period $P$, which means that the
vertical separation of drift bands of this component is coherent with
$P_{3}=3.0\pm 0.1\,P$. However, the fluctuation spectrum of the third pulse
component is diffused at the peaks of $3.0\pm 0.1\,P$ and 200$\pm$1 $P$ which
suggests that the vertical separation of drift bands is not fully coherent.
The asymmetric feature (along X=0) in the 2DFS of the first component suggests
the existence of pulse drifting across the pulse longitude. However, our
resolution is not enough to fully resolve the feature and yield a measurement
of $P_{2}$.

The $P_{3}$ for the first and third components seem identical. Correlation
analysis between them was carried out to check whether the intensity variation
following this period is phase-locked (more details see Kou et al. 2020). We
found no evidence that the intensity variations for the first and third pulse
components are phase-locked.

###  3.2 Pulse energy distribution

Figure 4: Normalised pulse energy distribution for PSR J0621+1002. The red,
black, green and blue lines are the pulse energies of off$-$pulse and
on$-$pulse region for the first and the third pulse components, respectively.
The dashed-dotted and dashed lines are the log-normal and power-law fitting
functions of the pulse energies for on-pulse region, respectively. The
vertical black line is the energy cutoff at 1.7 $E/\langle E\rangle$. Note
that the energy distribution is not deconvolved with the noise distribution.

Pulse energy distributions of the off-pulse and on-pulse regions for the first
and the third pulse components are presented in Figure 4. All the energies are
normalized by the mean on-pulse energy. On-pulse region is determined as the
longitude range where the pulse intensity significantly exceeds (larger than 3
$\sigma$) the baseline noise and the off-pulse energy is measured from a
region with the same duration as the on-pulse region in the baseline.

Pulse energy of the off-pulse region follows a narrow normal distribution,
while that of the on-pulse region can be described as log-normal distribution
with a high-energy power-law tail. The energy cutoff is at 1.7E/$\langle
E\rangle$ for the power-law tail (the vertical black line in Figure 4) and the
spectral index is $-8.0\pm 0.1$. Pulse energy distribution of the first pulse
component can be fitted well by a log-normal function. Pulse energy of the
third pulse component also follows a log-normal distribution with a power-law
tail. Generally, pulse energy for MSPs can be well modeled using log-normal or
Gaussian distribution (Shannon et al., 2014). However, log-normal energy
distribution with an excess of high-energy pulses has been detected in many
normal pulsars (Mickaliger et al., 2018).

###  3.3 The jitter noise

Figure 5: Single pulse timing residuals (black dots) for PSR J0621+1002. The
red bars are the ToA uncertainties.  Figure 6: Polarization profiles for the
mode A (left panel) and mode B (right panel) of PSR J0621+1002. The black,
red, and blue lines are the total intensity, linear polarized intensity, and
circular polarized intensity, respectively. Position angle (black dots) and
corresponding uncertainties (red bars) are shown as a function of pulse phase.
Figure 7: Estimates of jitter noise in PSR J0621+1002. Upper panel: variations
of rms timing residuals (squares) and ToA uncertainties (circles) versus the
number of pulses averaged. Bottom panel: quadrature difference (triangles)
between the rms timing residuals and ToA uncertainties. The red solid line is
the best fitting model for the jitter noise with index of $-0.50\pm 0.01$.
Figure 8:  Fraction of pulses used ($f$) versus normalised pulse jitter
($\sigma_{\rm J}(f,1)/\sigma_{\rm J}(1,1)$) for PSR J0621+1002. Circles from
right to left are the normalized pulse jitters of the pulses in class A and
class B, respectively. Squares from right to left are the normalized pulse
jitters of all the single pulses, single pulses with S/N less than 10, in the
range of 10 to 20, 20 to 30, 30 to 50, and larger than 50, respectively. The
red line is for the best fitting model for the jitter noise (see the red solid
line in Figure 7). Filled grey area identifies the region $\sigma_{\rm
J}(f,1)<\sigma_{\rm J}(1,1)/\sqrt{f}$, in which the improvement on timing
precision is achievable using sub-set of pulses with different S/N.

ToA uncertainties on short time-scales can be described as (Cordes & Shannon,
2010; Liu et al., 2012; Lam et al., 2016):

$\sigma^{2}_{\rm total}=\sigma^{2}_{\rm rm}+\sigma^{2}_{\rm J}+\sigma^{2}_{\rm
scint}+\sigma^{2}_{\rm 0},$ (1)

where $\sigma_{\rm rm}$, $\sigma_{\rm J}$, $\sigma_{\rm scint}$ and
$\sigma_{\rm 0}$ are the uncertainties induced by radiometer noise, jitter
noise, instability of short-term diffractive scintillation and all other
possible contributions. $\sigma_{\rm rm}$/$\sigma_{\rm J}$ is proportion to
the signal to noise ratio (S/N) of single pulse. For highly sensitive radio
telescopes, such as FAST (Liu et al., 2011, 2012; Hobbs et al., 2019),
$\sigma_{\rm rm}$ is expected to be less dominant especially for bright
pulsars.

ToA uncertainties induced by short-term diffractive scintillation can be
described as (see Cordes & Shannon (2010)): $\sigma_{\rm
scint}^{2}=\tau^{2}/N_{\rm scint}$ with the pulse-broadening time-scale $\tau$
and the number of scintles $N_{\rm scint}$ in the observation. The number of
scintles $N_{\rm scint}=(1+\eta\Delta\nu/\nu_{\rm d})(1+\eta\Delta T/t_{\rm
d})$, where $\Delta\nu$ and $\Delta T$ are the observing bandwidth and time,
$\nu_{\rm d}$ and $t_{\rm d}$ are the diffractive scintillation bandwidth and
time, and $\eta\approx 0.3$ is the scintillation filling factor (Cordes &
Shannon, 2010). For PSR J0621+1002, the diffractive scintillation bandwidth,
diffractive scintillation time and pulse-broadening time are 1.19 MHz, 323.4 s
and 0.16 $\mu$s (Cordes & Lazio, 2002), respectively. In our observation, the
observing bandwidth and time are 400 MHz and 1560 s, respectively; the
$\sigma_{\rm scint}$ is about 10 ns which is negligible.

Intrinsic single pulse shape and phase variations introduce jitter noise. We
carry out single pulse timing analysis for PSR J0621+1002 and the ephemeris is
provided by Desvignes et al. (2016). psrchive program paas was used to fit the
integrated profile of the entire observation and formed a noise-free template.
The ToAs are obtained by cross-correlating single pulse profiles with the
standard template. The timing residuals of single pulses are shown in Figure 5
and the post-fit timing residuals are divided into two classes. Single pulses
with timing residuals in the range of -5$-$4 ms and -10$-$-5 ms are defined as
class A and class B, respectively. The number of pulses in class A and class B
are 52429 and 1637, respectively.

The average pulse profiles of class A and class B are shown in Figure 6. It is
expected that the average profiles of the two classes are different. The pulse
profile of class A has a stronger third pulse component, and that of class B
has a stronger first pulse component. PAs of both classes show complicated
variations. However, there is no clear evidence that the PA swings of two
classes of pulses are different because of the limited signal to noise ratio
(S/N).

As mentioned above, $\sigma_{\rm scint}$ in Equation 1 is negligible. We can
assume that all of the excess error in the arrival time measurements are
attributed to jitter noise. Jitter noise can then be obtained by calculating
the quadrature difference between the observed rms timing residual and the ToA
uncertainty (Shannon et al., 2014): $\sigma_{\rm J}^{2}{(N_{\rm
p})}=\sigma_{\rm obs}^{2}{(N_{\rm p})}-\sigma_{\rm ToA}^{2}{(N_{\rm p})}$
where ${N_{\rm p}}$ is the number of averaged pulses. Estimates of jitter
noise of PSR J0621+1002 are shown in Figure 7. The red solid line in Figure 7
is the fitted result for the jitter noise scaling $\sigma_{\rm
J}{\rm(N_{p})}\propto N_{\rm p}^{-1/2}$. The expected jitter noise for one-
hour-long observation is 0.51 $\mu$s.

The expected timing precision for a fraction $f$ of $N$ pulses: $\sigma_{\rm
J}(f,N)=\sigma_{\rm J}(f,1)/\sqrt{fN}$ (Shannon et al., 2014), where
$\sigma_{\rm J}(f,N)$ is the jitter noise in the $fN$ selected single pulses.
While using a fraction of pulses, the timing precision improvement is achieved
if and only if $\sigma_{\rm J}(f,1)<\sigma_{\rm J}(1,1)\sqrt{f}$.

We examined the achievable timing precision using only the pulses in class A
or class B. The pulses in class A and class B and all the single pulses formed
three noise-free templates, respectively. ToAs for each class are obtained by
cross-correlating the single pulse profile with the corresponding standard
template. As shown in Figure 8, the $\sigma_{\rm J}$ does decrease if only the
pulses in one class are selected, but it does not decrease to where the timing
precision improvement can be achieved (the filled grey area in Figure 8).

We also examined the achievable timing precision for PSR J0621+1002 using only
bright pulses. All the single pulses are divided into five classes with the
S/N less than 10, in the range of 10 to 20, 20 to 30, 30 to 50, and larger
than 50. Note that the S/N is determined by dividing the peak flux density of
a single pulse by the rms of the off pulse region. The pulse numbers of these
five classes are 37185, 12050, 3022, 1476, and 333, respectively. We summed
all the single pulses in each class and formed five noise-free templates,
respectively. ToAs for each pulse class are obtained by cross-correlating the
single pulse profile with the corresponding standard template. As shown in
Figure 8, The $\sigma_{\rm J}$ does decrease for brighter pulses since they
originate from a narrower region of the pulse phase. However, timing precision
improvement is not achieved.

##  4  DISCUSSION AND CONCLUSIONS

PSR J0621+1002 shows periodic pulse intensity modulation for the first and
third pulse components. The horizontally collapsed fluctuation spectrum of the
first pulse component is peaked at 3.0$\pm$0.1 pulse periods, while that of
the third pulse component has two diffused peaks of 3.0$\pm$0.1 and 200$\pm$1
pulse periods. No modulation across the pulse phase was found. The periodic
emission variation has been detected in both normal pulsars and MSPs (Edwards,
& Stappers, 2003; Basu et al., 2016; Liu et al., 2016). By analyzing 70 normal
pulsars with periodic emission variations, Basu et al. (2020) suggested that
the physical origin of the periodic amplitude modulation is different from
that of the subpulse drifting. Generally, the subpulse drifting phenomenon is
only seen in the cone components of the pulse profile, while the periodic
amplitude modulation phenomenon is seen in the cone, even both the core and
cone profile components (Basu et al., 2019). The first and third components of
PSR J0621+1002 might associated with the cone emissions, which meets the
pattern of Basu et al. (2020).

Two classes of single pulses for PSR J0621+1002 have been distinguished and
the average pulse profiles for them are different. The duration of mode B is
only one or two pulse periods (about tens of milliseconds), while that of mode
change is typical in the range of several seconds to hours, even much longer
(Wang et al., 2007; Mahajan et al., 2018). The phenomenon seen in PSR
J0621+1002 is pulse jitter that due to radiation instability of single pulse,
while mode changing results from the change of pulsar magnetosphere geometries
or/and currents (Timokhin, 2010).

Jitter noise is currently thought to be a limiting noise process for sensitive
radio telescopes (Shannon et al., 2014; Hobbs et al., 2019). The jitter noise
level for PSR J0621+1002 is 0.51 $\mu$s for an hour-long observation. We
studied the achievable timing precision for this pulsar using the pules in
class A or class B, and the bright pulses and found that the timing precision
improvement is not achievable.

Similiar, by analyzing the single pulses of PSR J1713+0747, Liu et al. (2016)
found that no improvement of timing precision is achieved by selecting a sub-
set of pulses with specific flux density or pulse width. McKee et al. (2019)
studied the timing properties of giant pulses of PSR B1937+21 and found that
although the ToA uncertainties formed by giant pulses are much lower than that
of the average pulse profiles, the timing precision is not significantly
improved. However, the single pulses could still be used to mitigate jitter
noise for pulsars whose pulses with different properties or distributions in
phase. Kerr (2015) proposed a new method which can be used to reduce the
jitter noise associated with pulse-to-pulse variability.

## Acknowledgments

This work made use of the data from the Five-hundred-meter Aperture Spherical
radio Telescope, which is a Chinese national mega-science facility, operated
by National Astronomical Observatories, Chinese Academy of Sciences. The FAST
telescope is partly supported by The Operation, Maintenance and Upgrading Fund
for Astronomical Telescopes and Facility Instruments, budgeted from the
Ministry of Finance of China (MOF) and administrated by the Chinese Academy of
Science (CAS), and the Key Lab of FAST, National Astronomical Observatories,
Chinese Academy of Sciences. This work is supported by the National Natural
Science Foundation of China (No.12041304), the National SKA Program of China
(No.2020SKA0120100), the National Key Research and Development Program of
China (No.2017YFA0402600), the Youth Innovation Promotion Association of
Chinese Academy of Sciences and the 201* Project of Xinjiang Uygur Autonomous
Region of China for Flexibly Fetching in Upscale Talents.

## References

  * Arzoumanian et al. (2020) Arzoumanian, Z., Baker, P. T., Blumer, H., et al. 2020, ApJ, 905, L34.
  * Baars et al. (1977) Baars, J. W. M., Genzel, R., Pauliny-Toth, I. I. K., et al. 1977, A&A, 500, 135
  * Backer (1970a) Backer, D. C. 1970a, Nature, 228, 42
  * Backer (1970b) Backer, D. C. 1970b, Nature, 227, 692.
  * Backer (1976) Backer, D. C. 1976, ApJ, 209, 895.
  * Bartel & Hankins (1982) Bartel, N., & Hankins, T. H. 1982, ApJ, 254, L35
  * Basu et al. (2016) Basu, R., Mitra, D., Melikidze, G. I., et al. 2016, ApJ, 833, 29
  * Basu et al. (2019) Basu, R., Mitra, D., Melikidze, G. I., et al. 2019, MNRAS, 482, 3757
  * Basu et al. (2020) Basu, R., Mitra, D., & Melikidze, G. I. 2020, ApJ, 889, 133.
  * Camilo et al. (1996) Camilo, F., Nice, D. J., & Taylor, J. H. 1996, ApJ, 461, 812
  * Cordes & Shannon (2010) Cordes, J. M. & Shannon, R. M. 2010, arXiv:1010.3785
  * Cordes & Lazio (2002) Cordes, J. M. & Lazio, T. J. W. 2002, astro-ph/0207156
  * Desvignes et al. (2016) Desvignes, G., Caballero, R. N., Lentati, L., et al. 2016, MNRAS, 458, 3341
  * Edwards, & Stappers (2003) Edwards, R. T., & Stappers, B. W. 2003, A&A, 407, 273
  * Edwards & Stappers (2002) Edwards, R. T. & Stappers, B. W. 2002, A&A, 393, 733.
  * Hobbs et al. (2019) Hobbs, G., Dai, S., Manchester, R. N., et al. 2019, RAA, 19, 020
  * Hobbs et al. (2006) Hobbs, G. B., Edwards, R. T., & Manchester, R. N. 2006, MNRAS, 369, 655
  * Hotan et al. (2004) Hotan, A. W., van Straten, W., & Manchester, R. N. 2004, PASA, 21, 302
  * Jiang et al. (2019) Jiang, P., Yue, Y., Gan, H., et al. 2019, Science China Physics, Mechanics, and Astronomy, 62, 959502
  * Jenet et al. (2001) Jenet, F. A., Anderson, S. B., & Prince, T. A. 2001, ApJ, 546, 394
  * Jenet et al. (1998) Jenet, F. A., Anderson, S. B., Kaspi, V. M., et al. 1998, ApJ, 498, 365
  * Kasian (2012) Kasian, L. E. 2012, Ph.D. Thesis
  * Kerr (2015) Kerr, M. 2015, MNRAS, 452, 607.
  * Kou et al. (2020) Kou, F. F., Yan, W. M., Peng, B., et al. 2020, arXiv:2012.10156
  * Kramer et al. (1998) Kramer, M., Xilouris, K. M., Lorimer, D. R., et al. 1998, ApJ, 501, 270
  * Lam et al. (2019) Lam, M. T., McLaughlin, M. A., Arzoumanian, Z., et al. 2019, ApJ, 872, 193
  * Lam et al. (2016) Lam, M. T., Cordes, J. M., Chatterjee, S., et al. 2016, ApJ, 819, 155.
  * Liu et al. (2011) Liu, K., Verbiest, J. P. W., Kramer, M., et al. 2011, MNRAS, 417, 2916.
  * Liu et al. (2012) Liu, K., Keane, E. F., Lee, K. J., et al. 2012, MNRAS, 420, 361.
  * Liu et al. (2016) Liu, K., Bassa, C. G., Janssen, G. H., et al. 2016, MNRAS, 463, 3239
  * Liu et al. (2015) Liu, K., Karuppusamy, R., Lee, K. J., et al. 2015, MNRAS, 449, 1158
  * Lyne & Manchester (1988) Lyne, A. G. & Manchester, R. N. 1988, MNRAS, 234, 477.
  * Mahajan et al. (2018) Mahajan, N., van Kerkwijk, M. H., Main, R., et al. 2018, ApJ, 867, L2
  * Mickaliger et al. (2018) Mickaliger, M. B., McEwen, A. E., McLaughlin, M. A., et al. 2018, MNRAS, 479, 5413.
  * McKee et al. (2019) McKee, J. W., Stappers, B. W., Bassa, C. G., et al. 2019, MNRAS, 483, 4784
  * Rankin (1983) Rankin, J. M. 1983, ApJ, 274, 333.
  * Sobey et al. (2019) Sobey, C., Bilous, A. V., Grießmeier, J.-M., et al. 2019, MNRAS, 484, 3646.
  * Splaver et al. (2002) Splaver, E. M., Nice, D. J., Arzoumanian, Z., et al. 2002, ApJ, 581, 509
  * Shannon et al. (2013) Shannon, R. M., Ravi, V., Coles, W. A., et al. 2013, Science, 342, 334
  * Shannon et al. (2014) Shannon, R. M., Osłowski, S., Dai, S., et al. 2014, MNRAS, 443, 1463
  * Staelin & Reifenstein (1968) Staelin, D. H. & Reifenstein, E. C. 1968, Science, 162, 1481.
  * Timokhin (2010) Timokhin, A. N. 2010, MNRAS, 408, L41.
  * van Straten & Bailes (2011) van Straten, W. & Bailes, M. 2011, PASA, 28, 1.
  * Wang et al. (2020) Wang, S. Q., Hobbs, G., Wang, J. B., et al. 2020, ApJ, 902, L13.
  * Wang et al. (2007) Wang, N., Manchester, R. N., & Johnston, S. 2007, MNRAS, 377, 1383
  * Weltevrede (2016) Weltevrede, P. 2016, A&A, 590, A109.