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

# Global aspects of $3$-form gauge theory: implications for axion-Yang-Mills
systems

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

W

e investigate the proposition that axion-Yang-Mills systems are characterized
by a $3$-form gauge theory in the deep infrared regime. This hypothesis is
rigorously examined by initially developing a systematic framework for
analyzing $3$-form gauge theory coupled to an axion, specifically focusing on
its global properties. The theory consists of a BF term deformed by marginal
and irrelevant operators and describes a network of vacua separated by domain
walls converging at the junction of an axion string. It encompasses $(-1)$-,
$0$-, and $3$-form spontaneously broken global symmetries. Utilizing this
framework, in conjunction with effective field theory techniques and ’t Hooft
anomaly-matching conditions, we argue that the $3$-form gauge theory
faithfully captures the infrared physics of the axion-Yang-Mills system. The
ultraviolet theory is an $SU(N)$ Yang-Mills theory endowed with a massless
Dirac fermion coupled to a complex scalar and is characterized by chiral and
genuine $\mathbb{Z}_{m}^{(1)}$ 1-form center symmetries, with a mixed anomaly
between them. It features two scales: the vev of the complex scalar, $v$, and
the strong-coupling scale, $\Lambda$, with $\Lambda\ll v$. Below $v$, the
fermion decouples and two symmetries emerge: $U(1)^{(2)}$ $2$-form winding and
$\mathbb{Z}_{N}^{(1)}$ $1$-form center symmetries. As we flow below $\Lambda$,
matching the mixed anomaly necessitates introducing a dynamical $3$-form gauge
field of $U(1)^{(2)}$, which appears as the incarnation of a long-range tail
of the color field. The infrared theory possesses spontaneously broken chiral,
emergent $3$-form, and emergent $(-1)$-form global symmetries. It passes
several checks, among which: it displays the expected restructuring in the
hadronic sector upon transition between the vacua, and it is consistent under
the gauging of the genuine $\mathbb{Z}_{m}^{(1)}\subset\mathbb{Z}_{N}^{(1)}$
symmetry.

##  1 Introduction

Three-form gauge theory is a fascinating topic that attracted attention since
Lüscher’s seminal work Luscher:1978rn . There, it was argued that the non-
trivial topology of the $4$-D Yang-Mills theory shows up in the infrared as an
abelian long-range $3$-form gauge field $c_{3}$. It does not correspond to a
physical massless particle, i.e., a propagating degree of freedom.
Nevertheless, it contributes to the theory’s vacuum energy, i.e., cosmological
constant. In this formulation, the CP-violating $\theta$ term can be written
as

$\displaystyle\theta\int_{{\cal M}_{4}}dc_{3}\,,$ (1)

where $c_{3}$ is given in terms of the nonabelian Chern-Simons current
density:
$c_{3}=\mbox{tr}\left[\frac{1}{3}\left(a_{1}^{c}\right)^{3}+a_{1}^{c}\wedge
da_{1}^{c}\right]/(8\pi^{2})$, $a_{1}^{c}$ is the color field, and ${\cal
M}_{4}$ is the $4$-D manifold. The correlator of the derivatives of two Chern-
Simons current densities is the topological susceptibility $\chi$ of Yang-
Mills theory, which develops a pole as the momentum vanishes, the Kogut-
Susskind pole Kogut:1974kt , corresponding to a pole of the $c_{3}$
correlator. This is also known as the Veneziano ghost Veneziano:1979ec
because the pole appears with the opposite sign compared to those of
conventional particles, and it provides an alternative means (to Yang-Mills
instantons) to solve the axial $U(1)$ problem. One can write down an effective
action that reproduces these findings DiVecchia:1980yfw :

$\displaystyle S_{\scriptsize\mbox{IR}}=\frac{1}{2\chi}\int_{{\cal
M}_{4}}|dc_{3}|^{2}+\theta\int_{{\cal M}_{4}}dc_{3}\,,$ (2)

where a kinetic-energy term of $c_{3}$ is added to give the correct long-range
interaction form of the $c_{3}$ correlator. The effective action (2) yields
the vacuum energy Gabadadze:2002ff :

$\displaystyle E_{0}(\theta)=\frac{\chi}{2}\mbox{min}_{k}\left(\theta+2\pi
k\right)^{2}\,,\quad k\in\mathbb{Z}\,.$ (3)

This multi-branch function is periodic in $\theta$ and develops cusps at
$\theta=\pi,3\pi,...$, a result derived by Witten in the large-$N$ limit
Witten:1980sp . While this renowned result lends support to the validity of
(2), it is important to emphasize that the effective description (2) is
strictly derived in the large-$N$ limit. See Gabadadze:2002ff  for a review
and Aurilia:1980xj ; Duff:1980qv ; Freund:1980xh ; Nitta:2018yzb ;
Nitta:2018vyc  for supersymmetric extensions and further works on $3$-form
gauge theory.

When massless quarks are introduced, the $\theta$ term can be rotated away,
and the theory restores its CP invariance, thus solving the strong CP problem.
A similar effect can be achieved by introducing an axion $a$ through the
Peccei–Quinn mechanism Peccei:1977ur . This approach employs an anomalous
global $U(1)$ symmetry along with a complex scalar field whose phase
corresponds to the axion. The complex scalar undergoes spontaneous symmetry
breaking at a scale $v$, which is much higher than the strong-coupling scale
$\Lambda$. Below $\Lambda$, Yang-Mills instantons generate an effective
potential for $a+\theta$, which is minimized at a point in the field space
restoring the CP symmetry. The same conclusion can be reached using the
$3$-form gauge theory, as demonstrated by Dvali in Dvali:2005an ;
Dvali:2022fdv . In this framework, $c_{3}$ undergoes Higgsing when it absorbs
the axion, resulting in $c_{3}$ becoming a short-range field. This mechanism
can be elegantly seen in the Kalb-Ramond frame, where the axion is dualized to
a $2$-form gauge theory. This process effectively eliminates the second term
in (2) and restores the CP invariance.

Since $c_{3}$ does not carry a physical degree of freedom, it is reasonable to
question whether the $3$-form gauge theory is essential for formulating Yang-
Mills theory in the deep IR (without or with axions) or if it is merely a
redundant description lacking true physical significance. This work offers a
new perspective on the validity of the $3$-form gauge theory beyond the
large-$N$ limit. Under certain conditions, we argue that the $3$-form gauge
theory is a genuine and inevitable IR effective description in axion-Yang-
Mills systems. This is achieved by examining the global symmetries of such
systems as well as certain types of ’t Hooft anomaly-matching conditions.

Our understanding of symmetries has undergone a conceptual paradigm shift over
the past decade Gaiotto:2014kfa ; see Brennan:2023mmt  for a review and
Hidaka:2019mfm ; Brennan:2020ehu ; Yokokura:2022alv ; Yamamoto:2020vlk ;
Hidaka:2021kkf  for examples of works that discussed the $3$-form symmetries
from a modern perspective. In the contemporary paradigm, a $p$-form global
symmetry in $4$-D acts on a $p$-dimensional object and is generated by
operators (symmetry defects) living on $(3-p)$-dimensional topological
manifolds. Gauging a global symmetry is performed by introducing a background
field of the symmetry and including an arbitrary sum over inequivalent classes
of this background. Moreover, it has been realized recently that the concept
of symmetry can be extended to include operations that lack inversions; see,
e.g., Kaidi:2021xfk ; Cordova:2022ieu ; Choi:2022jqy ; Bhardwaj:2022lsg ;
Bartsch:2022mpm ; Anber:2023pny  and the reviews Shao:2023gho ; Schafer-
Nameki:2023jdn . Our primary goal is to conduct a systematic study of the
$3$-form gauge theory and apply it to Yang-Mills theory coupled to matter
fields and axions, with a focus on its global aspects. By utilizing newly
developed mathematical tools, we can thoroughly examine the $3$-form effective
field theory of axion-Yang-Mills systems. This effective description
successfully passes several consistency checks.

This work is divided into two main parts. The first part discusses the
symmetry aspects of a general low-energy $3$-form gauge theory coupled to
axion, including the multi-field case. Starting from a low-energy Lagrangian
exhibiting a $U(1)^{(2)}$ $2$-form along with $U(1)^{(0)}$ $0$-form global
symmetries, we construct a $3$-form gauge theory by gauging the former
symmetry. The resulting theory is a topological quantum field theory (TQFT) of
the BF type, modified by marginal and irrelevant operators, and describes a
set of $q$ domain walls ($q$ is a free parameter) separating $q$ distinct
vacua and forming a junction at the locus of an axion string. Generally, the
gauge theory exhibits a multitude of $\mathbb{Z}_{q}^{(0)}$ $0$-form,
$\mathbb{Z}_{q}^{(3)}$ $3$-form, and $U(1)^{(-1)}$ $(-1)$-form global
symmetries, with the first two symmetries exhibiting a mixed anomaly valued in
$\mathbb{Z}_{q}$. This anomaly is matched by breaking the two participating
symmetries, leading to $q$ distinct vacua separated by domain walls.
Furthermore, because $c_{3}$ is gapped in the presence of an axion, the
$U(1)^{(-1)}$ symmetry undergoes spontaneous breaking, which signifies the
absence of a cosmological constant. We also construct the symmetry defects
associated with these symmetries in two dual frames: the axion and the Kalb-
Ramond frames. We demonstrate the action of such symmetries within an example
of a domain-wall system. When the $3$-form symmetry or a subgroup thereof,
$\mathbb{Z}_{p}^{(3)}\subseteq\mathbb{Z}_{q}^{(3)}$, is gauged, the $0$-form
chiral symmetry becomes noninvertible, leading to the identification of some
of the theory’s vacua. This leaves $q/p$ distinct vacua.

In the second part of the paper, we apply this formalism to $SU(N)$ Yang-Mills
theory endowed with a single massless Dirac fermion in a representation ${\cal
R}$ coupled to a neutral complex scalar field. In the UV, this theory exhibits
both a $\mathbb{Z}_{2T_{\cal R}}^{\chi(0)}$ $0$-form chiral and
$\mathbb{Z}_{m}^{(1)}$ $1$-form center symmetries, with a possible mixed
anomaly between the two symmetries. Here, $T_{\cal R}$ is the Dynkin index of
${\cal R}$ and $m=\mbox{gcd}(N,n)$, where $n$ is the $N$-ality of ${\cal R}$.
This non-vanishing mixed anomaly, when matched below the strong-coupling
scale, necessitates the introduction of $c_{3}$. We summarize the idea here,
with details provided in the main body of the paper.

Let $v$ and $\Lambda$ be the complex scalar vev and the Yang-Mills strong-
coupling scales, respectively, and we take $\Lambda\ll v$. To see the
$\mathbb{Z}_{2T_{\cal R}}^{\chi(0)}-\mathbb{Z}_{m}^{(1)}$ mixed anomaly, we
turn on a $2$-form background field of $\mathbb{Z}_{m}^{(1)}$. This is
achieved, as in Kapustin:2014gua ; Gaiotto:2017yup ; Seiberg:2018ntt ;
Anber:2020xfk , by first introducing a $1$-form and $2$-form pair
$(B_{1}^{c},B_{2}^{c})$ along with the constraint $mB_{2}^{c}=dB_{1}^{c}$. The
field strength of the $1$-form field satisfies the quantization condition
$\int_{{\cal M}_{2}}dB_{1}^{c}\in 2\pi\mathbb{Z}$. Then the constraint implies
that $B_{2}^{c}$ has a vanishing field strength $dB_{2}^{c}=0$, while its
holonomy is fractional $\int_{{\cal
M}_{2}}B_{2}^{c}\in\frac{2\pi\mathbb{Z}}{m}$. To couple the background field
to fermions, in the second step, we enlarge $SU(N)$ to $U(N)$ and embed the
$\mathbb{Z}_{m}^{(1)}$ background field into the $U(1)$ part of $U(N)$.
However, we must ensure that the enlargement from $SU(N)$ to $U(N)$ does not
introduce new degrees of freedom, which can be done by postulating that the
theory is invariant under an auxiliary $1$-form gauge symmetry that acts
simultaneously on the $U(N)$ and $B_{2}^{c}$ fields. As we flow to energy
scales $\Lambda\ll E\ll v$, the magnitude of the complex scalar field freezes
at $v$, while the winding of its phase (the axion) leads to the emergence of a
$U(1)^{(2)}$ $2$-form symmetry that couples to the axion strings. Below
$\Lambda$, the strong dynamics set in, leading to the confinement of the color
field. Here, one is faced with a puzzle: the confinement of the color field
means that one should no longer incorporate the nonabelian field in the
calculations. If true, the low-energy theory is no longer invariant under the
postulated auxiliary $1$-form gauge symmetry, indicating that something is
missing. The way out is to introduce the dynamical $3$-form gauge field
$c_{3}$ of $U(1)^{(2)}$. The latter transforms non-trivially under the
auxiliary gauge symmetry, ensuring the full low-energy effective field theory
(in the background of the $\mathbb{Z}_{m}^{(1)}$ flux) is invariant under this
auxiliary symmetry. Moreover, the IR theory reproduces the mixed anomaly,
which is an important check on the analysis since anomalies are all-scale
phenomena. In this regard, $c_{3}$ can be thought of as the long-range tail of
the nonabelian dynamics. Even though it does not carry a physical degree of
freedom, its presence is essential for the consistency of the theory deep in
the IR.

Below the scale, $v$, the fermions become massive, with a mass of order $v$,
and decouple, leading to an emergent $\mathbb{Z}_{N}^{(1)}$ $1$-form symmetry.
The groups $U(1)^{(2)}$ and $\mathbb{Z}_{N}^{(1)}$ constitute a higher-group
structure, where the former is the parent and the latter is the daughter
symmetries. One may gauge the parent without gauging the daughter, but not
conversely. Therefore, as we flow below $\Lambda$, we may freely gauge
$U(1)^{(2)}$ without worrying about $\mathbb{Z}_{N}^{(1)}$, which stays an
emergent symmetry at such energy scales.

One of our main results is the IR effective field theory given by Eq. (5.1),
which we display here for convenience:

$\displaystyle{\cal L}_{E\ll\Lambda}=\frac{v^{2}}{2}da\wedge\star
da+\frac{T_{{\cal R}}a}{2\pi}\left(dc_{3}-\frac{N}{4\pi}B_{2}^{c}\wedge
B_{2}^{c}\right)+\Lambda^{4}{\cal
K}\left(\frac{dc_{3}-\frac{N}{4\pi}B_{2}^{c}\wedge
B_{2}^{c}}{\Lambda^{4}}\right)\,,$

where ${\cal K}$ is the kinetic energy term of $c_{3}$, and the background of
$\mathbb{Z}_{m}^{(1)}$ is activated. This theory exhibits exact
$U(1)^{(-1)}\times\mathbb{Z}_{T_{\cal R}}^{(0)}\times\mathbb{Z}_{T_{\cal
R}}^{(3)}$ symmetries. Dynamically, the IR theory forms axion domain walls,
separating $T_{\cal R}$ distinct minima and breaking $U(1)^{(-1)}$,
$\mathbb{Z}_{T_{\cal R}}^{(0)}$, and $\mathbb{Z}_{T_{\cal R}}^{(3)}$
maximally. The emergent $\mathbb{Z}_{N}^{(1)}$ symmetry is explicitly broken
by higher-order operators down to the genuine $\mathbb{Z}_{m}^{(1)}$ symmetry,
which remains intact.

The field strength of the $3$-form gauge field satisfies the quantization
condition $\int_{{\cal M}_{4}}dc_{3}=2\pi m$, where $m$ is an integer
equivalent to the topological charge of the Yang-Mills instantons. The full
partition function of the IR theory (at energy scale $E\ll\Lambda$) includes a
sum over all integers $m$. We may integrate out $c_{3}$, and using the Poisson
resummation formula, we obtain the Euclidean partition function

$\displaystyle Z[a]\sim\sum_{k\in\mathbb{Z}}e^{-i\frac{kN}{4\pi}\int_{{\cal
M}_{4}}B_{2}^{c}\wedge B_{2}^{c}}e^{-\int_{{\cal
M}_{4}}\frac{v^{2}}{2}da\wedge\star
da+\frac{\Lambda^{4}}{8\pi^{2}}\left(T_{\cal R}a+2\pi k\right)^{2}}\,.$ (5)

This partition function reproduces the chiral-center anomaly upon shifting
$a\rightarrow a+\frac{2\pi}{T_{\cal R}}$. It also displays an infinite number
of vacua, with the true vacuum energy given by

$\displaystyle V(a)\sim\Lambda^{4}\mbox{min}_{k}\left(T_{\cal R}a+2\pi
k\right)^{2}\,.$ (6)

The potential $V(a)$ has $T_{\cal R}$ minima at $2\pi\ell/T_{\cal R}$,
$\ell=0,1,...,T_{\cal R}-1$, as well as cusps at $a=\pi(2\ell+1)/T_{\cal R}$,
reflecting two facts. First, the cusps indicate that additional degrees of
freedom, not accounted for by $V(a)$, are sandwiched between the true minima
of the theory. These are the hadronic walls, which are very thin compared to
the thickness of the axion domain walls Fugleberg:1998kk ; Gabadadze:1999pp .
Second, a restructuring in the hadronic sector occurs as one goes between one
minimum and the other. These results are consistent with the large-$N$ limit
(3).

There exists a higher group structure between $\mathbb{Z}_{T_{\cal R}}^{(3)}$
and the emergent $\mathbb{Z}_{N}^{(1)}$ symmetries. However, this structure
trivialises for the genuine $\mathbb{Z}_{m}^{(1)}\subset\mathbb{Z}_{N}^{(1)}$
symmetry. This means we can gauge $\mathbb{Z}_{m}^{(1)}$ without worrying
about $\mathbb{Z}_{T_{\cal R}}^{(3)}$. Gauging the former gives
$SU(N)/\mathbb{Z}_{m}$ theory. This theory still exhibits a spontaneously
broken IR $\mathbb{Z}_{T_{\cal R}}^{(3)}$ symmetry due to the formation of
domain walls. However, the chiral symmetry $\mathbb{Z}_{T_{\cal R}}^{(0)}$
becomes noninvertible. This results in dressing the domain walls with an IR
TQFT. This intricate structure works as a consistency check on the adequacy of
using the $3$-form gauge theory to describe the axion-Yang-Mills systems’ IR
physics.

This paper is organized as follows. In Section 2, we set the stage by
considering the low-energy effective field theory of a compact scalar, which
possesses two global symmetries: shift and winding symmetries. The theory
encounters a mixed ’t Hooft anomaly, and thus, the shift symmetry breaks into
a discrete group upon gauging the winding symmetry. Next, we couple the gauge
field of the winding symmetry, the $3$-form gauge field, to the compact scalar
and analyze the resulting theory in great detail: we identify the global
symmetries, their mixed anomalies, and the noninvertible symmetries within
this theory. Section 3 is devoted to studying the compact scalar in the dual
frame, the Kalb-Ramond gauge theory, while Section 4 generalizes these
findings to two or more $3$-form gauge fields. In Section 5, we use the
machinery built in the previous sections to examine our proposal that the deep
IR regime of the axion-Yang-Mills systems is described by a $3$-form gauge
theory and apply various checks to this proposal. Finally, we conclude in
Section 6 with a brief discussion.

Throughout this paper, we use lowercase letters to denote dynamical fields,
i.e., fields we sum over in the path integral. Meanwhile, background fields
are denoted by uppercase letters. We mostly use the differential form notation
and subscripts to denote the degree of the form.

##  2 The Axion theory

We consider the $4$-D theory of a $2\pi$-periodic scalar field $a$, the axion,
i.e., we identify $a({\cal P})\equiv a({\cal P})+2\pi\mathbb{Z}$ at the
spacetime point ${\cal P}$. The basic Lagrangian is

$\mathcal{L}=\frac{v^{2}}{2}da\wedge\star da\,,$ (7)

where $v$ is a constant of mass dimension $1$. When coupling the axion to a
Yang-Mills theory via the Peccei-Quinn mechanism, $v$ is the axion’s symmetry-
breaking scale. The Lagrangian (7) has a global $U(1)^{(0)}$ 0-form shift
symmetry acting on the axion as $a\rightarrow a+\alpha$, where $\alpha$ is a
constant. The corresponding Noether’s $1$-form current is

$j_{1}=v^{2}da\,,$ (8)

which is conserved thanks to the equation of motion:

$d\star j_{1}^{(0)}=v^{2}d\star da=0\,.$ (9)

The topological symmetry generator (symmetry defect) enacting this
transformation is defined on a closed co-dimension-$1$ manifold
$\mathcal{M}_{3}$:

$U^{(0)}_{\alpha}(\mathcal{M}_{3})=e^{i\alpha\int_{\mathcal{M}_{3}}v^{2}\star
da}\,.$ (10)

The superscript emphasizes that the operator implements the action of a
$0$-form symmetry. We can take $U^{(0)}_{\alpha}(\mathcal{M}_{3})$ to surround
the local operator

$\displaystyle V({\cal M}_{0}={\cal P})=e^{ia({\cal P})}\,,$ (11)

and then topologically deforming $U^{(0)}_{\alpha}(\mathcal{M}_{3})$ past $V$
to find

$U^{(0)}_{\alpha}(\mathcal{M}_{3})V({\cal M}_{0})=e^{i\alpha}V({\cal
M}_{0})\,.$ (12)

The axion theory is also endowed with a $U(1)^{(2)}$ 2-form global symmetry
with a corresponding $3$-form current

$j_{3}=\star da\,,$ (13)

which is conserved because of the Bianchi identity:

$d\star j_{3}=d^{2}a=0\,.$ (14)

We can also define the symmetry defect of the $U(1)^{(2)}$ $2$-form symmetry
by integrating the Hodge-dual of $j_{3}$ on a co-dimension-$3$ manifold ${\cal
M}_{1}$ as:

$\displaystyle
U^{(2)}_{\beta}(\mathcal{M}_{1})=e^{i\beta\int_{\mathcal{M}_{1}}da}\,.$ (15)

This symmetry defect acts on the $2$-dimensional axion-string worldsheet
${\cal M}_{2}$ Brennan:2020ehu . Let $V(\mathcal{M}_{2})$ be the axion-string
Wilson surface, which has no local description in terms of the axion field
$a$. Deforming the symmetry defect $U^{(2)}_{\beta}(\mathcal{M}_{1})$ past
$V(\mathcal{M}_{2})$ transforms the latter by a phase:

$U^{(2)}_{\beta}(\mathcal{M}_{1})V(\mathcal{M}_{2})=e^{i\beta\text{Link}(\mathcal{M}_{1},\mathcal{M}_{2})}V(\mathcal{M}_{2})\,,$
(16)

where $\text{Link}(\mathcal{M}_{1},\mathcal{M}_{2})$ is the linking number
between the two manifolds.

There is a mixed ’t Hooft anomaly between $U(1)^{(0)}$ and $U(1)^{(2)}$
symmetries. To see it, we examine the commutation relation between the
symmetry defects $U^{(0)}_{\alpha}(\mathcal{M}_{3})$ and
$U^{(2)}_{\beta}(\mathcal{M}_{1})$. One way to perform the calculations is by
foliating ${\cal M}_{4}$ into constant time slices and orienting both ${\cal
M}_{3}$ and ${\cal M}_{1}$ to be time-like surfaces:

$\displaystyle U^{(0)}_{\alpha}(\mathcal{M}_{3}(t))=e^{i\alpha
v^{2}\int_{\mathcal{M}_{3}(t)}d^{3}x\partial_{0}a(x,t)}\,,\quad
U^{(2)}_{\beta}(\mathcal{M}_{1}(t))=e^{i\beta\int_{\mathcal{M}_{1}(t)}\partial_{i}adx^{i}}\,,$
(17)

where $i\in\\{1,2,3\\}$. Then, using the equal-time commutation relation
$\left[a(\bm{x},t),\Pi_{a}(\bm{y},t)\right]=i\delta^{(3)}(\bm{x}-\bm{y})$,
where $\Pi_{a}=v^{2}\partial_{0}a$, and employing the Baker-Campbell-Hausdorff
formula, we find

$\displaystyle
U^{(0)}_{\alpha}(\mathcal{M}_{3}(t))U^{(2)}_{\beta}(\mathcal{M}_{1}(t))=e^{-i\alpha\beta}U^{(2)}_{\beta}(\mathcal{M}_{1}(t))U^{(0)}_{\alpha}(\mathcal{M}_{3}(t))\,.$
(18)

The phase manifests the mixed anomaly: one cannot move the symmetry defects
freely without encountering non-trivial phases. The anomaly implies that
gauging one symmetry breaks the other into, at most, a discrete subgroup.

###  2.1 Gauging the $U(1)^{(2)}$ 2-form symmetry: $3$-form gauge theory and
domain walls

Now, we gauge the $U(1)^{(2)}$ symmetry, meaning that we introduce the
$3$-form gauge field $c_{3}$ of the $2$-form symmetry and perform the path
integral over $c_{3}$. We couple $c_{3}$ to its current $j_{3}$ by adding the
BF term

$\displaystyle\frac{q}{2\pi}\star j_{3}\wedge c_{3}=\frac{q}{2\pi}da\wedge
c_{3}$ (19)

to the Lagrangian (7). We introduced the positive integer $q\in\mathbb{N}$ as
a free parameter of the theory, and its physical significance will be apparent
below. The gauge field $c_{3}$ transforms as $c_{3}\rightarrow
c_{3}+d\lambda_{2}$ under the $U(1)^{(2)}$ gauge transformation, and via
integration by parts, we see that the new term (19) is invariant under this
transformation. The field strength of $c_{3}$ is $f_{4}=dc_{3}$, and it
satisfies the quantization condition:

$\displaystyle\int_{{\cal M}_{4}}f_{4}\in 2\pi\mathbb{Z}$ (20)

on a closed ${\cal M}_{4}$. The consistency of the theory under $U(1)^{(2)}$
large gauge transformations implies that $d\lambda_{2}$ satisfies the
condition

$\displaystyle\int_{{\cal M}_{3}}d\lambda_{2}\in 2\pi\mathbb{Z}\,.$ (21)

In addition, since $c_{3}$ is a dynamical field111The $3$-form gauge field has
mass dimension $3$., we can include a kinetic energy term ${\cal K}$ for
$c_{3}$. The total Lagrangian is222In a manifold with boundary, we also need
to consider a boundary term so that the variation of the kinetic term at the
boundary vanishes; see, e.g., Nitta:2018yzb  and references therein. We do not
run into this subtlety in this work.:

$\mathcal{L}=\frac{v^{2}}{2}da\wedge\star da-\frac{q}{2\pi}da\wedge
c_{3}+\Lambda^{4}{\cal K}\left(\frac{dc_{3}}{\Lambda^{4}}\right)\,,$ (22)

and we introduced the new scale $\Lambda$. More on ${\cal K}$ will be
discussed momentarily. The presence of the $c_{3}$ field reduces the
$U(1)^{(0)}$ symmetry to a $\mathbb{Z}^{(0)}_{q}\subset U(1)^{(0)}$ symmetry;
the Lagrangian (22) is only invariant under the shift $a\rightarrow a+2\pi/q$.
This demonstrates the earlier assertion that the mixed anomaly between the
$U(1)^{(2)}$ and $U(1)^{(0)}$ symmetries leads to the latter being broken into
a discrete subgroup when the former is gauged333As we shall see, this mixed
anomaly is the IR incarnation of the axial-color ABJ anomaly in an axion-Yang-
Mills UV complete theory.. The current conservation law (9) of the $0$-form
symmetry is modified to:

$v^{2}d\star da-\frac{q}{2\pi}dc_{3}=0\,.$ (23)

The corresponding $\mathbb{Z}_{q}^{(0)}$ symmetry defect is topological only
when we include this combination of fields - (10) is modified to

$U_{\alpha}^{(0)}(\mathcal{M}_{3})=e^{i\alpha\int_{\mathcal{M}_{3}}\left(v^{2}\star
da-\frac{q}{2\pi}c_{3}\right)}\,.$ (24)

Using the quantization condition on $d\lambda_{2}$, i.e., $\int_{{\cal
M}_{3}}d\lambda_{2}\in 2\pi\mathbb{Z}$, we readily see that
$U^{(0)}_{\alpha}(\mathcal{M}_{3})$ is gauge-invariant under a $U(1)^{(2)}$
gauge transformation if and only if $\alpha=2\pi\ell/q,\ell\in\mathbb{Z}$.
This seconds the above assertion that introducing the term (19) reduces the
$U(1)^{(0)}$ symmetry down to a $\mathbb{Z}_{q}^{(0)}$ subgroup.

Next, we focus on ${\cal K}$, the kinetic energy term of the $3$-form gauge
field. In the following, it will be helpful to write and analyze the
Lagrangian (22) in index notation444Translating from the $d$-forms to the
index notation, it helps to remember that we are working in Minkowski space,
with metric $\eta_{\mu\nu}=\mbox{diag}(+1,-1,-1,-1)$, such that
$\star\star=-1$ for even forms and $\star\star=+1$ for odd forms.:

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{v^{2}}{2}\partial_{\mu}a\,\partial^{\mu}a-\frac{q}{2\pi}\frac{1}{3!}\epsilon^{\mu\nu\rho\sigma}(\partial_{\mu}a)\,c_{\nu\rho\sigma}+\Lambda^{4}{\cal
K}\,,$ (25)

where $\epsilon^{\mu\nu\alpha\beta}$ is the Levi-Civita tensor and the Greek
indices run over $0,1,2,3$. The canonical (quadratic) form of the kinetic
energy term is:

$\displaystyle{\cal K}_{\scriptsize\mbox{can}}=-\frac{1}{2\cdot
4!\Lambda^{8}}f^{\mu\nu\alpha\beta}f_{\mu\nu\alpha\beta}\,,$ (26)

where
$f_{\mu\nu\alpha\beta}=\partial_{\mu}c_{\nu\alpha\beta}-\partial_{\nu}c_{\mu\alpha\beta}+\partial_{\alpha}c_{\mu\nu\beta}-\partial_{\beta}c_{\mu\nu\alpha}$.
Since $f_{\mu\nu\alpha\beta}$ is totally anti-symmetric in the $4$ indices, we
can always write it as

$\displaystyle f_{\mu\nu\alpha\beta}=-\epsilon_{\mu\nu\alpha\beta}f(x)\,,$
(27)

for some scalar function $f(x)$. Therfore, ${\cal K}_{\scriptsize\mbox{can}}$
takes the simple form555We used
$\epsilon_{\mu\nu\alpha\beta}\epsilon^{\mu\nu\alpha\beta}=-4!$. We can also
express $f$ in terms of $f_{\mu\nu\alpha\beta}$ as
$f=\frac{1}{4!}\epsilon^{\mu\nu\alpha\beta}f_{\mu\nu\alpha\beta}$.

$\displaystyle{\cal
K}_{\scriptsize\mbox{can}}=\frac{f^{2}(x)}{2\Lambda^{8}}\,.$ (28)

The mathematical statement (27) is equivalent to saying that the free $3$-form
field $c_{3}$ does not carry propagating degrees of freedom. To see that, use
the canonical kinetic term of $c_{3}$, ignore the axion in (25), and vary the
Lagrangian with respect to $c_{\mu\nu\alpha}$ to find

$\displaystyle\partial_{\mu}f^{\mu\nu\alpha\beta}=0\,,$ (29)

which admits the general solution
$f_{\mu\nu\alpha\beta}=-\epsilon_{\mu\nu\alpha\beta}f$, where $f$, in this
case, is a constant. The constant field strength of a $3$-form gauge field
carries no propagating degrees of freedom, much like free electrodynamics in
$2$-D. In the absence of $a$, the $4$-form field $f_{4}$ is a cosmological
constant, which is easily seen by substituting
$f_{\mu\nu\alpha\beta}=-\epsilon_{\mu\nu\alpha\beta}f$ into the
Lagrangian666However, a subtle issue arises because the cosmological constant
obtained in this manner does not align with the correct value derived by
varying (25) with respect to the metric tensor Nitta:2018vyc . This
discrepancy can be resolved by including boundary terms in (25). Nonetheless,
to obtain the correct form of the energy-momentum tensor and, consequently,
the cosmological constant, we will rely on the variation of the action with
respect to the metric tensor, disregarding boundary terms..

From the perspective of effective field theory, the kinetic energy term can
take a more generalized form, with ${\cal K}$ represented as a polynomial in
$f$:

$\displaystyle{\cal
K}\left(\frac{f}{\Lambda^{4}}\right)=\theta\frac{f}{\Lambda^{4}}+\frac{f^{2}}{2\Lambda^{8}}+c^{\prime}\frac{f^{4}}{\Lambda^{16}}+...\,,$
(30)

where $\theta$ and $c^{\prime}$ are some real parameters.The first term is
topological, and since $\int_{{\cal M}^{4}}f_{4}=\int_{{\cal M}_{4}}d^{4}xf\in
2\pi\mathbb{Z}$, the theory is invariant under the shift
$\theta\rightarrow\theta+2\pi$, and thus, $\theta$ is $2\pi$-periodic. This
term breaks the CP invariance unless $\theta=\\{0,\pi\\}$. However, $\theta$
can be rotated away by combining $\theta\frac{f}{\Lambda^{4}}$ with the second
term in (25), after integrating by parts and shifting $qa\rightarrow
qa-\theta$. Other possible higher-order kinetic energy terms may involve
higher derivatives of $f_{4}$. Yet, these terms may be plagued with ghosts
Nitta:2018vyc , and thus, we ignore such terms in our construction.

By rescaling the fields $a$ and $f$ as $a\rightarrow av$ and
$f\rightarrow\Lambda f$, we observe that the Lagrangian (25) manifests as a BF
theory deformed by marginal terms (the canonical kinetic terms of $a$ and $f$)
and by irrelevant operators (the higher-order terms of $f$). The full quantum
theory guarantees that the system has $q$-degenerate ground states777We thank
T. Sulejmanpasic and T. Tanizaki for discussions about this point.. To see
that, we may want to find an effective axion potential $V_{\text{eff}}(a)$ by
integrating out $f_{4}$ and imposing the constraint (20). To streamline the
analysis, we proceed by disregarding the axion kinetic energy term in (22),
which is a good approximation assuming $\Lambda\ll v$. Then, the Euclidean
partition function reads:

$\displaystyle Z[a]=\int[Dc_{3}]\sum_{m\in\mathbb{Z}}\delta\left(2\pi
m-\int_{{\cal M}_{4}}f_{4}\right)e^{-S_{E}}\,,$ (31)

and

$\displaystyle S_{E}=-\int_{{\cal M}_{4}}\Lambda^{4}{\cal
K}\left(if_{4}/\Lambda^{4}\right)+i\frac{q}{2\pi}a\wedge f_{4}\,.$ (32)

We can further simplify our analysis by recalling that $f_{4}$ does not carry
propagating degrees of freedom and can be expressed by Eq. (27). We take
${\cal M}_{4}$ to be a closed manifold, and therefore, we have $\int_{{\cal
M}_{4}}f_{4}=\int dV_{{\cal M}_{4}}f(x)=f_{0}V_{{\cal M}_{4}}$, where $f_{0}$
is the zero mode of $f(x)$ and $V_{{\cal M}_{4}}$ is the $4$-volume of ${\cal
M}_{4}$, and we assume $\Lambda^{4}V_{{\cal M}_{4}}\gg 1$. Focusing only on
the zero modes of $f$ and $a$, we find

$\displaystyle Z[a]\sim\left[\sum_{m\geq 0}e^{\Lambda^{4}V_{{\cal M}_{4}}{\cal
K}(2\pi im)}\cos(mqa_{0})\right]\times{\mbox{higher modes of}\,a}\,,$ (33)

where $a_{0}$ is the zero mode of $a$. The higher modes of $a$ are suppressed
by inverse powers of $\Lambda^{4}V_{{\cal M}_{4}}$ and can be neglected deep
in the IR. The effective potential is defined via $V_{\text{eff}}(a)=-V_{{\cal
M}_{4}}^{-1}\log Z[a]$, and thus, $V_{\text{eff}}(a_{0})\sim\Lambda^{4}{\cal
F}(qa_{0})$, where ${\cal F}$ is a periodic function with period $2\pi/q$. We
conclude that integrating out $f_{4}$ yields a periodic potential for the
axion that respects the $\mathbb{Z}_{q}^{(0)}$ shift symmetry, as it should
be. Minimizing $V_{\text{eff}}(a_{0})$ yields $q$-degenerate ground states
connected via domian walls.

An alternative way to perform the path integral in (31) is to use the Poisson
resummation formula $\sum_{m\in\mathbb{Z}}\delta\left(2\pi m-\int_{{\cal
M}_{4}}f_{4}\right)=\sum_{k\in\mathbb{Z}}e^{-ik\int_{{\cal M}_{4}}f_{4}}$.
Taking ${\cal K}$ in the canonical form ${\cal
K}_{\scriptsize\mbox{can}}=f_{4}^{2}/(2\Lambda^{2})$, focusing on the zero
modes, and performing the Gaussian integral, we obtain

$\displaystyle Z[a]\sim\sum_{k\in\mathbb{Z}}e^{-\frac{\Lambda^{4}V_{{\cal
M}_{4}}}{8\pi^{2}}\left(qa+2\pi k\right)^{2}}\,,$ (34)

which, again, is a periodic function that respects the $\mathbb{Z}_{q}^{(0)}$
shift symmetry. The result (34) is remarkable. This partition function
displays an infinite number of vacua, most of which are false. The true vacuum
energy is given by

$\displaystyle V(a)\sim\Lambda^{4}\mbox{min}_{k}\left(qa+2\pi k\right)^{2}\,.$
(35)

This potential has $q$ distinct minima, with cusps at $a=\pi/q$, $3\pi/q$,
etc. These cusps appear only upon including the infinite sum over $m$ in (33)
and performing the Poisson resummation formula. In other words, the cusps are
a feature of the full quantum theory. This observation will have far-reaching
consequences in axion-Yang-Mills systems.

Let us return to the Lagrangian (25) and study its classical aspects. Varying
it with respect to $c_{\nu\rho\sigma}$, assuming the general form of ${\cal
K}$, yields the equation of motion of the $3$-form field:

$\frac{q}{2\pi}\partial_{\mu}a=-\Lambda^{4}\partial_{\mu}{\cal
K}^{\prime}\left(\frac{f}{\Lambda^{4}}\right)\,,$ (36)

where the prime denotes the derivative with respect to the argument of ${\cal
K}$. Equation (36) is pivotal to our subsequent analysis. Integrating once, we
obtain

$\frac{q}{2\pi}(a-\tilde{a}_{0})=-\Lambda^{4}{\cal
K}^{\prime}\left(\frac{f}{\Lambda^{4}}\right)\,,$ (37)

for some integration constant $\tilde{a}_{0}$. Assuming ${\cal K}^{\prime}$ is
invertible, we can rearrange the equation of $f$:

$f=\Lambda^{4}({\cal
K}^{\prime})^{-1}\left(\frac{q}{2\pi\Lambda^{4}}(\tilde{a}_{0}-a)\right)\,.$
(38)

From (25), the equation of motion for the axion $a$ is

$v^{2}\partial_{\mu}\partial^{\mu}a-\frac{q}{2\pi}f=0\,.$ (39)

Here, $f$ acts as the derivative of a classical effective potential for the
axion field: in the presence of a classical effective potential for the axion,
the equation of motion is $v^{2}\partial_{\mu}\partial^{\mu}a+\frac{\partial
V_{\text{cl-eff}}(a)}{\partial a}=0$. This means we can set
$\frac{q}{2\pi}f=-\frac{\partial V_{\text{cl-eff}}(a)}{\partial a}$, and using
(38), we conclude

$\frac{\partial V_{\text{cl-eff}}(a)}{\partial
a}=-\frac{q}{2\pi}f=-\frac{q}{2\pi}\Lambda^{4}({\cal
K}^{\prime})^{-1}\left(\frac{q}{2\pi\Lambda^{4}}(\tilde{a}_{0}-a)\right)\,.$
(40)

Then, one can integrate (40) to obtain an expression of $V_{\text{cl-
eff}}(a)$. Unlike the effective potential obtained from the full partition
function, the classical effective potential does not need to yield $q$
degenerate ground states. The form of the classical effective potential
depends on the kinetic energy term for $c_{3}$. To elucidate this point, we
consider two examples of the resulting $V_{\text{cl-eff}}(a)$: the quadratic
and the cosine potentials. The corresponding kinetic energy functions are
Dvali:2005an :

$\displaystyle V_{\text{cl-eff}}^{\scriptsize\mbox{quadratic}}(a)$
$\displaystyle=$
$\displaystyle\frac{\Lambda^{4}}{2}(a-\tilde{a}_{0})^{2}\iff{\cal
K}_{\scriptsize\mbox{can}}\left(\frac{f}{\Lambda^{4}}\right)=\frac{q^{2}f^{2}}{2\Lambda^{8}}\,,$
(41)

and

$\displaystyle V_{\text{cl-eff}}^{\scriptsize\mbox{cos}}(a)$ $\displaystyle=$
$\displaystyle\Lambda^{4}(1-\cos\left(n(a-\tilde{a}_{0})\right)\iff$
$\displaystyle{\cal
K}_{\scriptsize\mbox{cos}}\left(\frac{f}{\Lambda^{4}}\right)$ $\displaystyle=$
$\displaystyle-1+\sqrt{1-\left(\frac{qf}{2\pi
n\Lambda^{4}}\right)^{2}}+\frac{qf}{2\pi
n\Lambda^{4}}\arcsin\left(\frac{qf}{2\pi n\Lambda^{4}}\right)\,.$ (42)

The integration constant in ${\cal K}$ is chosen such that ${\cal
K}\left(\frac{f}{\Lambda^{4}}=0\right)$=0.

The kinetic energy term (2.1) is designed to produce $V_{\text{cl-
eff}}^{\scriptsize\mbox{cos}}(a)$, and when $n\in q\mathbb{N}$, it exhibits
multiple-of-$q$ minima. Expanding ${\cal
K}_{\scriptsize\mbox{cos}}\left(\frac{f}{\Lambda^{4}}\right)$ to the leading
order in $f$ about one of these minima results, up to a proportionality
constant, in the canonical kinetic energy term (41). This, however, does not
imply that the canonical kinetic energy term fails to produce $q$-degenerate
ground states. As previously discussed, regardless of the form of ${\cal K}$,
the full partition function of the Lagrangian (22), incorporating the
quantization condition (20), will always lead to $q$-fold degeneracy in the
deep IR regime. Nonetheless, ${\cal
K}_{\scriptsize\mbox{cos}}\left(\frac{f}{\Lambda^{4}}\right)$ proves
invaluable as it facilitates connections with textbook examples of axion
domain walls. One can think of it as a UV completion of the canonical kinetic
energy.

Domin walls. In the following, we proceed to discuss the classical domain wall
solutions in the $U(1)^{(2)}$ gauge theory. We shall use the effective cosine
potential in (2.1) to carry out our analysis. However, this section’s
conclusions also hold for arbitrary potential, i.e., for arbitrary forms of
${\cal K}$.

The cosine potential yields $n$ vacua $a_{\ell}=\frac{2\pi\ell}{n}$, where
$\ell=0,1,,...,n-1$. There are $n$ domain walls separating the adjacent vacua
with a kink-like profile given by (here, we may set $\tilde{a}_{0}=0$):

$\displaystyle
a(z)=\frac{2\pi\ell}{n}+\frac{4}{n}\mbox{arctan}\left(e^{nm_{a}z}\right)\,,\quad-\infty<z<\infty\,,$
(43)

and $m_{a}=\frac{\Lambda^{2}}{v}$. We assumed the walls are space-filling in
the $x$ and $y$ directions with a profile along the $z$-direction, taking
${\cal M}_{4}=\mathbb{R}^{4}$ for simplicity. We also assumed that the walls
are separated by distances much larger than their width $\sim m_{a}^{-1}$. In
the following, the statement $z\rightarrow\pm\infty$ means that $|z|\gg
m_{a}^{-1}$ but still away from the adjacent walls. We observe that the
reality of the kinetic energy term ${\cal K}$, see (2.1), implies the
inequality $|f|\leq 2\pi n\Lambda^{4}/q$. The value of ${\cal K}$ attains its
minimum value, ${\cal K}=1$, at $f=0$, while it is maximized at $|f|=2\pi
n\Lambda^{4}/q$. In addition, the first equality in (40) yields:

$\displaystyle f(z)$ $\displaystyle=$ $\displaystyle-\frac{2\pi
n}{q}\Lambda^{4}\sin\left(na(z)\right)=-\frac{2\pi
n}{q}\Lambda^{4}\sin\left[4\mbox{arctan}\left(e^{nm_{a}z}\right)\right]\,.$
(44)

The theory has a $\mathbb{Z}_{q}^{(0)}$ $0$-form symmetry that acts as
$a\rightarrow a+\frac{2\pi}{q}$. The invariance of the cosine potential under
the $0$-form symmetry demands that

$\displaystyle\frac{n}{q}\in\mathbb{N}\,,$ (45)

and thus $n\geq q$. Later on, we shall discuss that a stack of $n$ domain
walls intersect on a string carrying a charge $q$ under the $U(1)^{(2)}$ gauge
symmetry. However, it is more energetically favorable for the charge-$q$
string to support only $n=q$ domain walls. Nevertheless, we maintain the
generality of $n$ and $q$ in our subsequent discussion888For example, we could
have a kinetic energy term ${\cal K}$ that corresponds to a more general form
of the effective potential $V_{\text{cl-eff}}(a)=\sum_{m\geq
1}\Lambda_{m}^{4}\left(1-\cos(mqa)\right)$..

In the following, the derivatives of ${\cal K}$ will prove useful for our
study:

$\displaystyle{\cal K}^{\prime}=\frac{q}{2\pi
n\Lambda^{4}}\mbox{arcsin}\left(\frac{qf}{2\pi
n\Lambda^{4}}\right)\,,\quad{\cal K}^{\prime\prime}=\frac{q^{2}}{(2\pi
n\Lambda^{4})^{2}\sqrt{1-\left(\frac{qf}{2\pi n\Lambda^{4}}\right)^{2}}}\,.$
(46)

Also, two important limiting behaviours of $f(z)$ are worth noticing:

$\displaystyle f(z\rightarrow 0)$ $\displaystyle=$ $\displaystyle\frac{4\pi
n^{2}}{q}z\Lambda^{4}\,,$ $\displaystyle f(z\rightarrow\pm\infty)$
$\displaystyle=$ $\displaystyle\frac{8\pi
n}{q}\Lambda^{4}e^{-nm_{a}|z|}\mbox{sign}(z)\,.$ (47)

At the wall core, $z=0$, we find that $f$ attains its mimimum value $f(0)=0$,
where ${\cal K}$ is also minimized, while in the vacum
$z\rightarrow\pm\infty$, we similarly have $f(\pm\infty)=0$, where again
${\cal K}$ is minimized. However, there exists a distance inside the wall,
$\pm|z_{m}|$, at which $|f(z_{m})|=\frac{2\pi n}{q}\Lambda^{4}$, i.e., it is
maximized:

$\displaystyle m_{a}|z_{m}|=\frac{1}{n}\log\tan\left(\frac{\pi}{8}\right)\,.$
(48)

At $|z_{m}|$, the kinetic energy ${\cal K}$ is also maximized. Thus, $|f(z)|$
monotonically increases from the core of the domain wall until it reaches
$z_{m}$, after which it starts decreasing exponentially; see Figure 1. The
exponential decay observed in $f(z)$ is a defining trait of a gapped system.
In this scenario, the $4$-form field $f_{4}$ eats the axion, resulting in its
acquisition of mass. This is also evident from

$\displaystyle\int_{\mathbb{R}^{4}}f_{4}=0\,,$ (49)

a result consistent with the quantization condition (20). In contrast, in a
gapless system, $|f(z)|$ would remain at the constant value of $\frac{2\pi
n}{q}\Lambda^{4}$ from $\pm z_{m}$ to infinity. Below, we shall show that
$\frac{q}{2\pi}\left(a(\infty)-a(-\infty)\right)=q/n$ is the domain wall
charge under a $\mathbb{Z}_{q}^{(3)}$ $3$-form global symmetry. Thus, the
stack of $n$ domain walls carries a total charge of $q$.

Figure 1: The profiles of $f$ (solid line) and ${\cal
K}_{\scriptsize\mbox{cos}}$ (dashed line) as functions of the coordinate $z$
(not to scale). We take $q=n=2$ and set $\Lambda=1$.

###  2.2 The $3$\- and $(-1)$-form global symmetries, anomalies, and
noninvertible symmetry

In this section, we show that the pure $U(1)^{(2)}$ gauge theory possesses a
$U(1)^{(3)}$ $3$-form global symmetry, which undergoes a breakdown into a
$\mathbb{Z}^{(3)}_{q}$ symmetry in the presence of charge-$q$ matter. In
addition, the theory is endowed with a $(-1)$-form symmetry that can be used
to diagnose the existence of a cosmological constant. We further demonstrate
that there is a mixed ’t Hooft anomaly between $\mathbb{Z}^{(3)}_{q}$ and
$\mathbb{Z}_{q}^{(0)}$ symmetries. Gauging the former gives rise to a
$U(1)^{(2)}/\mathbb{Z}_{q}$ gauge theory.

$3$-form global symmetry. To begin, we rewrite Eq. (36), the equation of
motion of $c_{3}$, in vacuum, i.e., setting the left-hand-side to $0$, as

$\displaystyle{\cal
K}^{\prime\prime}\left(\frac{f}{\Lambda^{2}}\right)\partial_{\mu}f^{\mu\nu\alpha\beta}=0\,,$
(50)

which implies either ${\cal K}^{\prime\prime}=0$ or
$\partial_{\mu}f^{\mu\nu\alpha\beta}=0$. The first equation holds only when
${\cal K}$ is extremized, and thus, we concentrate solely on the latter
equation that can be rewritten in the $d$-form language as

$\displaystyle d\star dc_{3}=0\,.$ (51)

This takes the form of the conservation law of the Hodge dual of a $4$-form
current:

$\displaystyle d\star j_{4}=0\,,\quad\star j_{4}=\frac{\star
dc_{3}}{\Lambda^{4}}=\frac{\star f_{4}}{\Lambda^{4}}\,,$ (52)

implying that gauging the $U(1)^{(2)}$ $2$-form symmetry in vacuum gives rise
to an emergent $U(1)^{(3)}$ $3$-form global symmetry. The $3$-form symmetry
couples to $3$-surfaces ${\cal M}_{3}$, such that the Wilson surface operator
is given by

$\displaystyle V({\cal M}_{3})=e^{ip\int_{{\cal M}_{3}}c_{3}}\,,\quad
p\in\mathbb{Z}\,.$ (53)

The value $p=1$ gives the fundamental Wilson surface, while values of $p>1$
are higher representations. The conserved charge of this symmetry $Q^{(3)}$ is
given by integrating $\star j_{4}$ over a $0$-dimensional manifold, or in
other words, it is the local operator $\star j_{4}({\cal P})=\star f_{4}({\cal
P})/\Lambda^{4}$ at the spacetime point ${\cal P}$. Using $\star
f_{4}=\frac{1}{4!}\epsilon^{\mu\nu\alpha\beta}f_{\mu\nu\alpha\beta}$ along
with $f_{\mu\nu\alpha\beta}=-\epsilon_{\mu\nu\alpha\beta}f$ and
$\epsilon^{\mu\nu\alpha\beta}\epsilon_{\mu\nu\alpha\beta}=-4!$, we find

$\displaystyle Q^{(3)}=\star j_{4}({\cal P})=\frac{f({\cal
P})}{\Lambda^{4}}\,,$ (54)

and, thus, the generator of the symmetry (symmetry defect) is

$\displaystyle U_{\gamma}^{(3)}({\cal M}_{0}={\cal P})=e^{i\gamma\star
j_{4}}=e^{i\gamma f/\Lambda^{4}}\,,$ (55)

where $\gamma\in[0,2\pi)$. The symmetry defect measures the amount of flux
carried by a Wilson surface. Upon pushing $U_{\gamma}^{(3)}({\cal M}_{0})$
past $V({\cal M}_{3})$, we obtain the algebra:

$\displaystyle U_{\gamma}^{(3)}({\cal M}_{0})V({\cal
M}_{3})=e^{ip\gamma\text{Link}(\mathcal{M}_{0},\mathcal{M}_{3})}V({\cal
M}_{3})\,.$ (56)

Let us repeat the analysis in the presence of matter. We shall use two
approaches. First, we will investigate the $3$-form symmetry in the vicinity
of the wall but far enough from its core. Next, we shall redo the analysis,
this time without making any approximations or assumptions about the nature of
the wall. We start with the approximate method, analyzing the situation near
the domain walls of the cosine potential we studied above. We assume that we
are far from the domain wall core, i.e., we are considering distances
$|z|>z_{m}$, where $z_{m}$ is given by (48), which is the distance at which
$f$ and ${\cal K}$ are maximized. At $z_{m}$, ${\cal K}^{\prime\prime}$ is
ill-defined, invalidating our analysis; this is why we need to perform the
calculations far from the core. Keeping this constraint in mind, we start by
rewriting Eq. (36) in the $d$-form language as:

$\displaystyle\Lambda^{4}{\cal K}^{\prime\prime}(f)d\star
dc_{3}=-\frac{q}{2\pi}da\,.$ (57)

Far from the core, we can safely set $f\cong 0$, as $f(z)$ decays
exponentially fast at distances $z>z_{m}$. We have999Consider if we had
employed the quadratic potential, as represented by Eq. (41). In such a
scenario, we would obtain ${\cal K}^{\prime\prime}=q^{2}/\Lambda^{8}$. The
disparity of $(2\pi n)^{2}$ between these two cases becomes evident when we
recognize that the function $f$ reaches its zero precisely at the local
minimum of ${\cal K}$ in the case of the cosine potential, where we may
approximate ${\cal K}$ by a quadratic function ${\cal K}\cong
1+\frac{q^{2}f^{2}}{2(2\pi n)^{2}\Lambda^{8}}$. ${\cal
K}^{\prime\prime}(f\cong 0)=q^{2}/(4\pi^{2}n^{2}\Lambda^{8})$. Using this
information, we can rearrange (57) as a conservation law:

$\displaystyle d\star j_{4}=0\,,\quad Q^{(3)}=\star j_{4}({\cal P})$
$\displaystyle=$ $\displaystyle\frac{q^{2}}{(2\pi n)^{2}\Lambda^{2}}\star
f_{4}({\cal P})+\frac{q}{2\pi}a({\cal P})$ (58) $\displaystyle=$
$\displaystyle\frac{q^{2}}{(2\pi n)^{2}\Lambda^{2}}f({\cal
P})+\frac{q}{2\pi}a({\cal P})\,.$

The symmetry defect is

$\displaystyle U_{\gamma}^{(3)}({\cal M}_{0}={\cal P})=e^{i\gamma\star
j_{4}}=e^{i\gamma\left(\frac{q^{2}f(\cal P)}{(2\pi
n)^{2}\Lambda^{4}}+\frac{qa({\cal P})}{2\pi}\right)}\,.$ (59)

Since $a({\cal P})$ and $a({\cal P})+2\pi$ are identified,
$U_{\gamma}^{(3)}({\cal M}_{0}={\cal P})$ is meaningful only when $\gamma\in
2\pi\mathbb{Z}/q\mathbb{Z}\equiv 2\pi\mathbb{Z}_{q}$. The charged object under
the global symmetry is still the $3$-dimensional Wilson surface $V({\cal
M}_{3})$ given by (53). Now, the algebra of $U_{\ell}^{(3)}({\cal M}_{0})$,
$\ell\in\mathbb{Z}_{q}$, and $V({\cal M}_{3})$ is given by:

$\displaystyle U_{\ell}^{(3)}({\cal M}_{0})V({\cal M}_{3})=e^{i\frac{2\pi
p\ell}{q}\text{Link}(\mathcal{M}_{0},\mathcal{M}_{3})}V({\cal M}_{3})\,.$ (60)

As we emphasized above, expression (59) is valid only far from the domain wall
core, i.e., (59) is consistent with setting $f\cong 0$. However, inspired by
the preceding analysis, we can repeat the treatment without making any
approximations or assumptions about the nature of the walls. Our central
equation, as usual, is (36), which we will write as a conservation law101010In
the cosine potential example, the derivative of $Q^{(3)}$ is ill-defined at
the core. Nevertheless, $Q^{(3)}$ is well-defined everywhere.:

$\displaystyle\partial_{\mu}Q^{(3)}=0\,,\quad Q^{(3)}({\cal
P})=\frac{q}{2\pi}a({\cal P})+\Lambda^{4}{\cal K}^{\prime}\left(\frac{f({}\cal
P)}{\Lambda^{4}}\right)\,,$ (61)

with corresponding symmetry defect

$\displaystyle U_{\ell}^{(3)}({\cal
M}_{0})=e^{i\frac{2\pi\ell}{q}\left(\frac{q}{2\pi}a({\cal P})+\Lambda^{4}{\cal
K}^{\prime}\left(\frac{f({}\cal
P)}{\Lambda^{4}}\right)\right)}\,,\quad\ell=1,2,..,q\,.$ (62)

This is the generator of a $\mathbb{Z}_{q}^{(3)}$ symmetry for a generic form
of the kinetic energy ${\cal K}$. It is easy to check that (61) reproduces the
approximate expression (59) of the cosine potential near $f\cong 0$. The
charge $Q^{(3)}({\cal P})$ is a constant of motion unless one encounters a
domain wall: crossing an elementary wall changes $Q^{(3)}$ by

$\displaystyle\Delta
Q^{(3)}=\frac{q}{2\pi}\left(a(\infty)-a(-\infty)\right)=\frac{q}{n}$ (63)

units. In other words, $\Delta Q^{(3)}$ is the domain wall charge under the
$\mathbb{Z}_{q}^{(3)}$ global symmetry. Interestingly, when $n=q$, the most
natural scenario, $\Delta Q^{(3)}$ coincides with the concept of topological
charge in the theory of solitons. As we shall discuss in Section 3, the domain
walls intersect in a line; this is the locus of a string. From the flux
conservation, this string carries a charge $q$, evenly distributed among the
$n$ intersecting domain walls. We conclude that there are $n$ dynamical walls
attached to a string, carrying a total charge $Q^{(3)}=q$ under
$\mathbb{Z}_{q}^{(3)}$.

The $3$-form symmetry $\mathbb{Z}_{q}^{(3)}$ acts on $q$ distinct Wilson
surfaces $V({\cal M}_{3})=e^{ip\int_{{\cal M}_{3}}c_{3}}$, $p=1,2,..,q$. When
$p\neq 0$ Mod $q$, these Wilson surfaces are topological and cannot end on the
dynamical walls since the latter always come in a stack of a total charge $q$.

It remains to discuss the fate of the $3$-form global symmetry. We start with
the $U(1)^{(3)}$ symmetry in the absence of matter, i.e., taking
$v\rightarrow\infty$. In this case, as we discussed before, the equation of
motion of the $3$-form gauge field $c_{3}$ yields a constant solution:
$f_{\mu\nu\alpha\beta}=-\epsilon_{\mu\nu\alpha\beta}f$, where $f$ is a
constant. Two walls experience a constant force, meaning that the $U(1)^{(3)}$
is unbroken and the Wilson surface $V({\cal M}_{3})$ exhibits the “area” law
$\langle V({\cal M}_{3})\rangle=0$. Introducing the axion field and the
coupling $-\frac{q}{2\pi}da\wedge c_{3}$ breaks $U(1)^{(3)}$ down to
$\mathbb{Z}_{q}^{(3)}$ and endows the theory with a $\mathbb{Z}_{q}^{(0)}$
$0$-form global symmetry. Now, the $4$-form field $f_{4}$ is gapped, and the
Wilson surfaces exhibit the “perimeter” law $\langle V({\cal
M}_{3})\rangle\neq 0$, meaning that $\mathbb{Z}_{q}^{(3)}$ is spontaneously
broken. Moreover, if the theory forms domain walls, $\mathbb{Z}_{q}^{(0)}$
also breaks spontaneously.

The $(-1)$-form symmetry and the cosmological constant. In addition, the
theory possess a $U(1)$ $(-1)$-form symmetry. The Bianchi identity
$d^{2}c_{3}=0$ can be written as the conservation law of the Hodeg-dual of a
“magnetic” current $\star j_{4}^{m}=dc_{3}=f_{4}$. The corresponding symmetry
defect is

$\displaystyle U^{(-1)}_{\gamma}\left({\cal
M}_{4}\right)=e^{i\gamma\int_{{\cal M}_{4}}f_{4}}\,,\quad\mbox{noting
that}\quad\int_{{\cal M}_{4}}f_{4}\in 2\pi\mathbb{Z}\,.$ (64)

There are no physical objects that are acted upon with
$U^{(-1)}_{\gamma}\left({\cal M}_{4}\right)$. Yet, the $(-1)$-form symmetry is
a diagnosis for a gapped/gapless $f_{4}$ Aloni:2024jpb . In the domain wall
example discussed above, we found $\int_{\mathbb{R}^{4}}f_{4}=0$ (see Eq.
(49)), meaning $\langle
U^{(-1)}_{\gamma}\left(\mathbb{R}^{4}\right)\rangle=1$, and thus, the
$(-1)$-form symmetry is spontaneously broken. This is expected since $f_{4}$
is Higgsed (gapped) by the axion.

The $(-1)$-form symmetry is intricately linked to the cosmological constant.
The energy-momentum tensor of $c_{3}$ can be derived directly from
(22)111111Remember that we use the boundary condition ${\cal K}(f=0)=0$.:

$\displaystyle T_{\mu\nu}=\eta_{\mu\nu}\Lambda^{4}\left[{\cal
K}\left(\frac{f}{\Lambda^{4}}\right)-\frac{f}{\Lambda^{4}}{\cal
K}^{\prime}\left(\frac{f}{\Lambda^{4}}\right)\right]\,,$ (65)

which takes the form of a cosmological constant. In a gapped system, $f=0$
(this is true far from the domain wall core); thus, $T_{\mu\nu}=0$. Therefore,
a theory with a spontaneously broken $(-1)$-form symmetry does not have a
cosmological constant. Conversely, an unbroken $(-1)$-form symmetry signifies
the presence of a cosmological constant.

The mixed ’t Hooft anomaly between $\mathbb{Z}_{q}^{(3)}$ and
$\mathbb{Z}_{q}^{(0)}$ symmetries. An important question is whether there is a
mixed anomaly between the two global symmetries $\mathbb{Z}_{q}^{(3)}$ and
$\mathbb{Z}_{q}^{(0)}$. To answer this question, we examine the commutation
relation between $U_{\ell}^{(3)}({\cal M}_{0})$ and
$U_{\ell^{\prime}}^{(0)}(\mathcal{M}_{3})$ given by (62) and (24),
respectively. The calculations can be performed, as before, by foliating
${\cal M}_{4}$ into constant time slices and orienting ${\cal M}_{3}$ to be a
time-like surface:

$U_{\ell^{\prime}}^{(0)}(\mathcal{M}_{3}(t))=e^{i\frac{2\pi\ell^{\prime}}{q}\int_{\mathcal{M}_{3}(t)}\left(v^{2}\partial_{0}a-\frac{q}{2\pi}\frac{c_{ijk}\epsilon^{ijk}}{3!}\right)}\,.$
(66)

Then, using the equal-time commutation relation
$\left[a(\bm{x},t),\Pi_{a}(\bm{y},t)\right]=i\delta^{(3)}(\bm{x}-\bm{y})$,
where
$\Pi_{a}=v^{2}\partial_{0}a-\frac{q}{2\pi}\frac{c_{ijk}\epsilon^{ijk}}{3!}$,
we obtain

$\displaystyle U_{\ell^{\prime}}^{(0)}(\mathcal{M}_{3}(t))U_{\ell}^{(3)}({\cal
M}_{0}(t))=e^{i\frac{2\pi\ell\ell^{\prime}}{q}}U_{\ell}^{(3)}({\cal
M}_{0}(t))U_{\ell^{\prime}}^{(0)}(\mathcal{M}_{3}(t))\,.$ (67)

Thus, a $\mathbb{Z}_{q}$-valued mixed anomaly exists between the two
symmetries. The anomaly implies one or both symmetries are broken. Our
previous discussion reveals that both symmetries are broken in a theory that
forms domain walls in the IR.

Gauging $\mathbb{Z}_{q}^{(3)}$ and the noninvertible
$\tilde{\mathbb{Z}_{q}}^{(0)}$. Next, we consider the
$U(1)^{(2)}/\mathbb{Z}_{p}$ gauge theory resulting from gauging a subgroup
$\mathbb{Z}_{p}^{(3)}\subseteq\mathbb{Z}_{q}^{(3)}$. Gauging the discrete
symmetry $\mathbb{Z}_{p}^{(3)}$ means introducing a background gauge field of
the symmetry and including a sum over arbitrary insertions of this background
in the path integral. We can introduce the $\mathbb{Z}_{p}^{(3)}$ background
$F_{4}$ into the path integral by replacing every $f_{4}$ in (22) by
$f_{4}\rightarrow f_{4}+F_{4}$, where $F_{4}$ can be expressed as
$pF_{4}=dF_{3}$ and $dF_{3}$ satisfies the quantization condition $\int_{{\cal
M}_{4}}dF_{3}\in 2\pi\mathbb{Z}$. In turn, this implies the quantization of
$F_{4}$ in units of $1/p$, i.e., $\int_{{\cal
M}_{4}}F_{4}\in\frac{2\pi}{p}\mathbb{Z}$, such that $dF_{4}$=0.

Now consider the second term in (22) in the presence of the
$\mathbb{Z}_{q}^{(3)}$ background. Its Euclidean form is
$i\frac{q}{2\pi}a\wedge\left(f_{4}+F_{4}\right)$. Since, $\int_{{\cal
M}_{4}}F_{4}\in\frac{2\pi}{p}\mathbb{Z}$, only the shift $a\rightarrow
a+\frac{2\pi p}{q}$, i.e., $\mathbb{Z}^{(0)}_{q/p}$, survives as a genuine
(invertible) discrete symmetry. Actually, gauging $\mathbb{Z}_{p}^{(3)}$
renders $\tilde{\mathbb{Z}}_{q}^{(0)}$ as a non-invertible symmetry, within
which its subgroup $\mathbb{Z}^{(0)}_{q/p}$ remains invertible. To see this,
consider the commutation relation between the $\mathbb{Z}_{q}^{(0)}$ symmetry
defect $U_{\ell}^{(0)}(\mathcal{M}_{3})$ and the generator of the
$\mathbb{Z}_{p}^{(3)}$ symmetry $\left[U_{1}^{(3)}({\cal M}_{0})\right]^{q/p}$
in the original $U(1)^{(2)}$ gauge theory. From (67) we have

$\displaystyle\left[U_{1}^{(3)}(\mathcal{M}_{0})\right]^{-q/p}U_{\ell}^{(0)}({\cal
M}_{3})\left[U_{1}^{(3)}(\mathcal{M}_{0})\right]^{q/p}=e^{i\frac{2\pi\ell}{p}}U_{\ell}^{(0)}({\cal
M}_{3})\,.$ (68)

This relation shows that $U_{\ell}^{(0)}({\cal M}_{3})$ fails to be a gauge-
invariant operator in the $U(1)^{(2)}/\mathbb{Z}_{p}$ gauge theory. To remedy
the problem, we sum over an arbitrary number of gauge transformations under
the operator $\left[U_{1}^{(3)}(\mathcal{M}_{0})\right]^{q/p}$ by defining the
new $U(1)^{(2)}/\mathbb{Z}_{p}$ gauge-invariant symmetry defect121212We use a
tilde to denote the noninvertible symmetries and their generators.:

$\displaystyle\tilde{U}_{\ell}^{(0)}({\cal M}_{3})$ $\displaystyle\equiv$
$\displaystyle\sum_{r\in\mathbb{Z}}\left[U_{1}^{(3)}(\mathcal{M}_{0})\right]^{-rq/p}U_{\ell}^{(0)}({\cal
M}_{3})\left[U_{1}^{(3)}(\mathcal{M}_{0})\right]^{rq/p}$ (69) $\displaystyle=$
$\displaystyle U_{\ell}^{(0)}({\cal
M}_{3})\sum_{r\in\mathbb{Z}}e^{i\frac{2\pi\ell r}{p}}$ $\displaystyle=$
$\displaystyle U_{\ell}^{(0)}({\cal M}_{3})\delta_{\ell\in p\mathbb{Z}}\,.$

The second line (69) implies that $\tilde{U}_{\ell}^{(0)}({\cal M}_{3})$ is
nonunitary:

$\displaystyle\tilde{U}_{\ell}^{(0)}({\cal
M}_{3})\tilde{U}_{\ell}^{(0)}(\overline{{\cal
M}_{3}})=\sum_{r,r^{\prime}\in\mathbb{Z}}e^{i\frac{2\pi\ell(r-r^{\prime})}{p}}\neq
1\,,$ (70)

where $\overline{{\cal M}_{3}}$ is the orientation reversal of ${\cal M}_{3}$.
Thus, the generator of the $\tilde{\mathbb{Z}}_{q}^{(0)}$ symmetry is not
unitary; gauging $\mathbb{Z}_{p}^{(3)}$ renders $\tilde{\mathbb{Z}}_{q}^{(0)}$
a noninvertible symmetry. The third line in (69) shows that
$\tilde{U}_{\ell}^{(0)}({\cal M}_{3})$ is a projective operator: recalling
$\ell=0,1,..,q$, we conclude that the subgroup
$\mathbb{Z}_{q/p}^{(0)}\subset\tilde{\mathbb{Z}}_{q}^{(0)}$ survives as an
invertible symmetry131313We thank E. Poppitz for discussion about this point..

The $U(1)^{(2)}/\mathbb{Z}_{p}$ gauge theory has a remaining
$\mathbb{Z}_{q/p}^{(3)}$ global symmetry with symmetry defect given by

$\displaystyle\left[U_{\ell}^{(3)}({\cal M}_{0})\right]^{p}=e^{i\frac{2\pi\ell
p}{q}\left(\frac{q}{2\pi}a({\cal P})+\Lambda^{4}{\cal
K}^{\prime}\left(\frac{f({}\cal
P)}{\Lambda^{4}}\right)\right)}\,,\quad\ell=1,2,..,q/p.$ (71)

The $\mathbb{Z}_{q/p}^{(3)}$ symmetry acts on the $U(1)^{(2)}/\mathbb{Z}_{p}$
gauge-invariant Wilson surfaces $V({\cal M}_{3})=e^{imp\int_{{\cal
M}_{3}}c_{3}}$, $m\in\mathbb{Z}_{q/p}$. The action of
$\mathbb{Z}_{q/p}^{(0)}\subset\tilde{\mathbb{Z}}_{q}^{(0)}$ takes us between
the $q/p$ vacua of the theory. When we fully gauge $\mathbb{Z}_{q}^{(3)}$,
i.e., in $U(1)^{(2)}/\mathbb{Z}_{q}$ gauge theory, we have neither genuine
$0$-form nor $3$-form global symmetries.

What has just happened, especially concerning a generic kinetic energy term
${\cal K}$, that leads to the formation of dynamical domain walls? Let us
analyze the situation by considering the cosine potential (2.1), setting $n=q$
for simplicity. We shall also gauge the full $\mathbb{Z}_{q}^{(3)}$ symmetry.
In this case, what we are operationally doing is that we are declaring the
equivalence between all the $q$ vacua: $a\equiv a+\frac{2\pi\ell}{q}$,
$\ell=1,2,..,q$. Therefore, it is more meaningful to define $\varphi\equiv qa$
and replace $V_{\scriptsize\mbox{eff}}(a)=\Lambda^{4}\left(1-\cos(qa)\right)$
with
$V_{\scriptsize\mbox{eff}}(\varphi)=\Lambda^{4}\left(1-\cos\varphi\right)$.
The latter potential supports a single domain wall interpolating between
$\varphi=0$ and $\varphi=2\pi$. However, such a wall is unstable quantum
mechanically as it decays by instanton effects. We end up with a theory with a
unique vacuum, supporting only strings but no domain walls.

##  3 The dual description: the Kalb-Ramond frame

While the axion framework provided valuable insights into the global
symmetries within our system, the notion of strings remained implicit. To
address this, we transition to the Kalb-Ramond frame Kalb:1974yc , where the
presence and properties of strings become more evident and accessible.

To dualize the axion Lagrangian (7) to a theory of a $2$-form Kalb-Ramond
field, we add to the Lagrangian (7) an extra term Dvali:2005an :

$\mathcal{L}=\frac{v^{2}}{2}da\wedge\star da-\frac{1}{2\pi}b_{2}\wedge
d^{2}a\,.$ (72)

Here, $b_{2}$ is a Lagrange multiplier used to impose the Bianchi
identity141414The form of the extra term implies that $b_{2}$ has mass
dimension $2$.\- integrating out $b_{2}$ gives $d^{2}a=0$. We can also
integrate out $a$ via its equation of motion:

$2\pi v^{2}\star da=db_{2}\,.$ (73)

Substituting back into (72) gives the dual Kalb-Ramond theory:

$\mathcal{L}_{\text{dual}}=\frac{1}{2(2\pi)^{2}v^{2}}db_{2}\wedge\star
db_{2}\,.$ (74)

The Kalb-Ramond field $b_{2}$ couples to the $2$-dimensional axion-string
worldsheet $\mathcal{M}_{2}$. Thus, the Wilson-like operator of an axion-
string is:

$V({\mathcal{M}_{2}})=e^{i\int_{\mathcal{M}_{2}}b_{2}}\,.$ (75)

Unlike $V({\mathcal{M}_{2}})$, the operator $V({\cal M}_{0})$, which in the
original theory was given by $e^{ia}$, has no local description in terms of
$b_{2}$. Therefore, in the Kalb-Ramond frame, $V({\mathcal{M}_{2}})$ and
$V({\cal M}_{0})$ behave respectively like Wilson and ’t Hooft operators in
electrodynamics. This picture is reversed in the axion frame.

In the dual description, the $0$-form and $2$-form global symmetry currents
are:

$j_{1}=\frac{1}{2\pi}\star db_{2},\qquad j_{3}=2\pi v^{2}db_{2}\,.$ (76)

These currents satisfy the conservation laws:

$\displaystyle d\star j_{1}=0\,,\quad d\star j_{3}=0\,,$ (77)

which are the results of the Bianchi identity $d^{2}b_{2}=0$ and the equation
of motion $d\star db_{2}=0$, respectively. The corresponding $U(1)^{(0)}$ and
$U(1)^{(2)}$ symmetry defects are given by

$U^{(0)}_{\alpha}(\mathcal{M}_{3})=e^{i\alpha\int_{\mathcal{M}_{3}}\frac{1}{2\pi}db_{2}}\,,\quad
U^{(2)}_{\beta}(\mathcal{M}_{1})=e^{i\beta\int_{\mathcal{M}_{1}}2\pi
v^{2}\star db_{2}}\,.$ (78)

With this dual formulation, we can see the action of $U(1)^{(2)}$ global
symmetry- it shifts $b_{2}$ by a constant 2-form $\Lambda_{2}$:

$b_{2}\rightarrow b_{2}+\Lambda_{2}\,,$ (79)

where $b_{2}$ has charge $q$ under $U(1)^{(2)}$. This transforms the axion
string by a $U(1)$ phase:

$U_{\beta}^{(2)}(\mathcal{M}_{2})V({\mathcal{M}_{2}})=e^{i\beta\text{Link}(\mathcal{M}_{1},\mathcal{M}_{2})}V({\mathcal{M}_{2}})\,.$
(80)

###  3.1 Gauging the $U(1)^{(2)}$ $2$-form symmetry

Here, we derive the Lagrangian of the dual Kalb-Ramond gauge theory, which
results by gauging the $U(1)^{(2)}$ symmetry. Our starting point is the
Lagrangian (22) after adding the term $-b_{2}\wedge d^{2}a/(2\pi)$ to enforce
the Bianchi identity $d^{2}a=0$:

$\mathcal{L}=\frac{v^{2}}{2}da\wedge\star da-\frac{q}{2\pi}da\wedge
c_{3}-\frac{1}{2\pi}b_{2}\wedge d^{2}a+\Lambda^{4}{\cal
K}\left(\frac{dc_{3}}{\Lambda^{4}}\right)\,.$ (81)

The equation of motion of $a$ is:

$\displaystyle d\star da-\frac{1}{2\pi}d^{2}b_{2}-\frac{q}{2\pi}dc_{3}=0\,,$
(82)

and integrating once we find

$\displaystyle\star da=\frac{1}{2\pi v^{2}}\left(db_{2}+qc_{3}\right)\,.$ (83)

Substituting (83) into (81), we obtain the dual Lagrangian

$\displaystyle\mathcal{L}_{\text{dual}}=\frac{1}{2(2\pi)^{2}v^{2}}(db_{2}+qc_{3})\wedge\star(db_{2}+qc_{3})+\Lambda^{4}{\cal
K}\left(\frac{dc_{3}}{\Lambda^{4}}\right)\,.$ (84)

In this formulation, the $U(1)^{(2)}$ symmetry is gauged by introducing the
$3$-form gauge field $c_{3}$, which couples minimally to the Kalb-Ramond field
$b_{2}$. The minimal coupling $db_{2}+qc_{3}$ means that $b_{2}$ carries a
charge $q$ under $c_{3}$. This is also manifest in the fact that the dual
Lagrangian is invariant under the $U(1)^{(2)}$ local gauge transformation
$c_{3}\rightarrow c_{3}+d\lambda_{2}$, $b_{2}\rightarrow b_{2}-q\lambda_{2}$.
We may think of $b_{2}$ as the Stuckelberg field of $c_{3}$; as $c_{3}$ eats
up the $b_{2}$ field, it aquires a mass $\sim\frac{\Lambda^{2}}{v}$. In the
limit $v\rightarrow\infty$, the Kalb-Ramond field decouples, leaving us with a
pure $U(1)^{(2)}$ gauge theory.

The effect of gauging the $U(1)^{(2)}$ symmetry is that the spectrum of
extended operator changes. The Wlison surface operator (75) is no longer
gauge-invariant under the $2$-form gauge symmetry. In the Kalb-Ramond frame, a
gauge-invariant operator is

$e^{i\int_{\mathcal{M}_{3}}db_{2}+qc_{3}}=e^{i\int_{{\cal
M}_{2}=\partial\mathcal{M}_{3}}b_{2}}e^{iq\int_{\mathcal{M}_{3}}c_{3}}\,,$
(85)

which can be interpreted as a string attached to a stack of domain walls with
a cumulative charge of $q$ (the charge under the $c_{3}$ field). An elementary
axion that winds around this configuration cannot detect a nontrivial phase.
This can be envisaged by computing the commutator

$\displaystyle\left[e^{ia},e^{i\int_{\mathcal{M}_{3}}db_{2}+qc_{3}}\right]=0\,,$
(86)

where we used (83) along with
$\left[a(\bm{x},t),\Pi_{a}(\bm{y},t)\right]=i\delta^{(3)}(\bm{x}-\bm{y})$ and
$\Pi_{a}=v^{2}\partial_{0}a-\frac{q}{2\pi}\frac{c_{ijk}\epsilon^{ijk}}{3!}$.

We end this section by discussing the global symmetries in the Kalb-Ramond
frame. First, the generator of the $\mathbb{Z}_{q}^{(0)}$ $0$-form symmetry is
given by $U^{(0)}_{\alpha}(\mathcal{M}_{3})$ in (78). This generator, however,
must be invariant under a $U(1)^{(2)}$ gauge transformation $b_{2}\rightarrow
b_{2}-q\lambda_{2}$. Using $\int_{{\cal M}_{3}}d\lambda_{2}\in
2\pi\mathbb{Z}$, we find $\alpha=\frac{2\pi\ell}{q}$, $\ell=1,2,..,q$, as
expected for the $\mathbb{Z}_{q}^{(0)}$ symmetry. Second, we also have a
$\mathbb{Z}_{q}^{(3)}$ $3$-form global symmetry. However, the generator of
this symmetry has no local description in the Kalb-Ramond frame. It is crucial
to highlight that transitioning between the axion and Kalb-Ramond frames does
not eliminate global symmetries. Rather, certain symmetries may not be
manifest in a local description. Finally, the $(-1)$-form symmetry is still
given by (64).

##  4 Multi $3$-form gauge theory

Consider an axion $a$ coupled to $2$ distinct $3$-form gauge fields $c_{3}$
and $\tilde{c}_{3}$. The Lagrangian reads

$\mathcal{L}=\frac{v^{2}}{2}|da|^{2}-\frac{q_{1}}{2\pi}da\wedge
c_{3}-\frac{q_{2}}{2\pi}da\wedge\tilde{c}_{3}+\Lambda^{4}{\cal
K}\left(\frac{f_{4}}{\Lambda^{4}}\right)+\Lambda^{4}\tilde{{\cal
K}}\left(\frac{\tilde{f}_{4}}{\Lambda^{4}}\right)\,,$ (87)

where $f_{4}=dc_{3}$ and $\tilde{f}_{4}=d\tilde{c}_{3}$ are the field
strengths of $c_{3}$ and $\tilde{c}_{3}$, respectively, and we assumed that
the scale $\Lambda$ is the same for all the $3$-form fields. Notice that we
did not include a kinetic-mixing term to avoid complications. The $3$-form
gauge fields $c_{3}$ and $\tilde{c}_{3}$ are invariant under $U(1)^{(2)}\times
U(1)^{(2)}$ gauge transformations $c_{3}\rightarrow c_{3}+d\lambda_{2}$ and
$\tilde{c}_{3}\rightarrow\tilde{c}_{3}+d\tilde{\lambda}_{2}$, where
$\int_{{\cal M}_{3}}d\lambda_{2},\int_{{\cal M}_{3}}d\tilde{\lambda}_{2}\in
2\pi\mathbb{Z}$, while the field strengths satisfy the quantization conditions

$\displaystyle\int_{{\cal M}_{4}}f_{4}\,,\int_{{\cal M}_{4}}\tilde{f}_{4}\in
2\pi\mathbb{Z}\,.$ (88)

The equations of motion of $a$, $c_{3}$, and $\tilde{c}_{3}$ read

$\displaystyle v^{2}d\star
da-\left(\frac{q_{1}}{2\pi}dc_{3}+\frac{q_{2}}{2\pi}d\tilde{c}_{3}\right)=0\,,\quad\frac{q_{1}}{2\pi}\partial_{\mu}a=-\Lambda^{4}\partial_{\mu}{\cal
K}^{\prime}\left(\frac{f}{\Lambda^{4}}\right)\,,$
$\displaystyle\frac{q_{2}}{2\pi}\partial_{\mu}a=-\Lambda^{4}\partial_{\mu}\tilde{\cal
K}^{\prime}\left(\frac{\tilde{f}}{\Lambda^{4}}\right)\,.$ (89)

Trading $f$ and $\tilde{f}$ for a classical axion effective potential yields

$\displaystyle\frac{\partial V_{\text{cl-eff}}(a)}{\partial
a}=-\frac{1}{2\pi}\left(q_{1}f+q_{2}\tilde{f}\right)\,.$ (90)

This relationship asserts that the combination of fields
$q_{1}f+q_{2}\tilde{f}$ vanishes at the extrema of $V_{\text{cl-eff}}(a)$.
This implies that this particular combination of the $3$-form fields is gapped
at the theory’s vacua. Conversely, the independent combination
$q_{2}f-q_{1}\tilde{f}$ remains ungapped Dvali:2005an .

The system (87) enjoys a multitude of global symmetries. First, the theory is
invariant under a $\mathbb{Z}_{q}^{(0)}$ symmetry that acts on $a$ as
$a\rightarrow a+\frac{2\pi}{q}$ and $q=\mbox{gcd}(q_{1},q_{2})$. The generator
of $\mathbb{Z}_{q}^{(0)}$ is

$U_{\ell}^{(0)}(\mathcal{M}_{3})=e^{i\frac{2\pi\ell}{q}\int_{\mathcal{M}_{3}}\left(v^{2}\star
da-\frac{q_{1}c_{3}+q_{2}\tilde{c}_{3}}{2\pi}\right)}\,,\quad\ell=1,2,...,q\,,$
(91)

which acts on the local operator $e^{ia({\cal P})}$.

In addition, the system exhibits two independent $3$-form global symmetries.
To find them, we use two methods. We start with the first method, which was
not discussed previously but works as an alternative view on the global
$3$-form symmetry. In this method, we shift $c_{3}$ and $\tilde{c}_{3}$ by two
independent closed but not exact 3-forms $\Lambda_{3}$ and
$\tilde{\Lambda}_{3}$:

$c_{3}\rightarrow
c_{3}+\Lambda_{3}\,,\quad\tilde{c}_{3}\rightarrow\tilde{c}_{3}+\tilde{\Lambda}_{3}\,,$
(92)

under which the action gets shifted by

$S\rightarrow S+\frac{q_{1}}{2\pi}\int_{{\cal
M}_{4}}da\wedge\Lambda_{3}+\frac{q_{2}}{2\pi}\int_{{\cal
M}_{4}}da\wedge\tilde{\Lambda}_{3}=S+q_{1}k\alpha+q_{2}k\beta\,,$ (93)

where we defined $\int_{{\cal M}_{3}}\Lambda_{3}=\alpha,\int_{{\cal
M}_{3}}\tilde{\Lambda}_{3}=\beta$, and we recalled that $\int_{{\cal
M}_{1}}da=2\pi k,k\in\mathbb{Z}$ since $a$ is a compact scalar. Under the
$\Lambda_{3}$ and $\tilde{\Lambda}_{3}$ shifts, the path integral picks up a
phase:

$Z\rightarrow e^{iq_{1}k\alpha+iq_{2}k\beta}Z\,.$ (94)

It is easily seen that there are two combinations of $\alpha$ and $\beta$ that
lead to two independent $3$-form global symmetries that leave the action
invariant: a $U(1)^{(3)}$ symmetry is obtained by setting
$q_{1}\alpha=-q_{2}\beta$ and a $\mathbb{Z}_{q}^{(3)}$ symmetry is obtained
upon taking $\alpha,\beta\in\frac{2\pi}{q}\mathbb{Z}$, where
$q=\gcd(q_{1},q_{2})$. These are linearly independent transformations, so
there are no redundancies, and the faithful 3-form symmetry group is

$\mathbb{Z}_{q}^{(3)}\times U(1)^{(3)}\,.$ (95)

Another way to obtain the same result is by combining the equations of motion
of $c_{3}$ and $\tilde{c}_{3}$ in the form of two independent conservation
laws. Using (4) we find

$\displaystyle\partial_{\mu}\left(\frac{q_{1}a+q_{2}a}{2\pi}+\Lambda^{4}{\cal
K}^{\prime}\left(\frac{f}{\Lambda}\right)+\Lambda^{4}\tilde{\cal
K}^{\prime}\left(\frac{\tilde{f}}{\Lambda}\right)\right)$ $\displaystyle=$
$\displaystyle 0\,,$ $\displaystyle\partial_{\mu}\left(-q_{2}\Lambda^{4}{\cal
K}^{\prime}\left(\frac{f}{\Lambda}\right)+q_{1}\Lambda^{4}\tilde{\cal
K}^{\prime}\left(\frac{\tilde{f}}{\Lambda}\right)\right)$ $\displaystyle=$
$\displaystyle 0\,,$ (96)

from which we define the two symmetry defects:

$\displaystyle U_{\alpha_{1}}^{(3)}\left({\cal M}_{0}\right)$ $\displaystyle=$
$\displaystyle e^{i\alpha_{1}\left(\frac{q_{1}a+q_{2}a}{2\pi}+\Lambda^{4}{\cal
K}^{\prime}\left(\frac{f}{\Lambda}\right)+\Lambda^{4}\tilde{\cal
K}^{\prime}\left(\frac{\tilde{f}}{\Lambda}\right)\right)}\,,\quad
U_{\alpha_{2}}^{(3)}\left({\cal
M}_{0}\right)=e^{i\alpha_{2}\left(-q_{2}\Lambda^{4}{\cal
K}^{\prime}\left(\frac{f}{\Lambda}\right)+q_{1}\Lambda^{4}\tilde{\cal
K}^{\prime}\left(\frac{\tilde{f}}{\Lambda}\right)\right)}\,.$

While the phase $\alpha_{2}$ is an arbitrary $U(1)$ phase, implying that
$U_{\alpha_{2}}^{(3)}$ is the symmetry defec of a $U(1)^{(3)}$ $3$-form global
symmetry, the single-valuedness of $U_{\alpha_{1}}^{(3)}$ as $a\sim a+2\pi$
implies that $\alpha_{1}=\frac{2\pi\mathbb{Z}}{q}$, reducing the second
symmetry group from $U(1)^{(3)}$ down to $\mathbb{Z}_{q}^{(3)}$, in accordance
with our earlier finding.

The theory also possesses two distinct $U(1)^{(-1)}$ $(-1)$-form symmetries
associated with the Bianchi’s identities: $d^{2}c_{3}=d^{2}\tilde{c}_{3}=0$.
As we mentioned above, only the field combination
$q_{1}c_{3}+q_{2}\tilde{c}_{3}$ is gapped while the other combination
$-q_{2}c_{3}+q_{1}\tilde{c}_{3}$ remains gappless. This implies that only one
of the two $(-1)$-form symmetries is spontaneously broken while the other
remains intact, and thus, the theory with a single axion and two $3$-form
gauge fields possesses a cosmological constant in the deep IR.

This treatment is easily generalized to any $K$ distinct $3$-form fields to
find that the full faithful symmetry group is

$\mathbb{Z}_{q}^{(0)}\times\mathbb{Z}_{q}^{(3)}\times\prod_{i=1}^{K-1}U(1)^{(i)(3)}$
(98)

where $q=\gcd(q_{1},\ldots,q_{K})$.

##  5 UV completion: axion-Yang-Mills theory

In this section, we argue that the $3$-form gauge theory in either the axion
or the Kalb-Ramond frame emerge in the IR from a UV-complete axion-Yang-Mills
system. This conclusion is reached by using effective field theory methods
empowered by new ’t Hooft anomaly matching conditions. We put the IR effective
field theory under scrutiny by testing its adequacy under various checks.

To this end, consider an $SU(N)$ gauge theory endowed with a massless Dirac
fermion in a representation ${\cal R}$ under $SU(N)$. In addition, consider a
complex scalar $\Phi$ that is inert under $SU(N)$ but otherwise couples to the
Dirac fermion. The Lagrangian of the system reads Anber:2020xfk :

$\displaystyle{\cal L}=-\frac{1}{2g^{2}}\mbox{tr}\left(f_{2}^{c}\wedge\star
f_{2}^{c}\right)+\bar{\psi}\bar{\sigma}^{\mu}D_{\mu}\psi+\bar{\tilde{\psi}}\bar{\sigma}^{\mu}D_{\mu}{\tilde{\psi}}+|d\Phi|^{2}-V(\Phi)+y\Phi\tilde{\psi}\psi+\mbox{h.c.}\,.$
(99)

$f_{2}^{c}=da_{1}^{c}-ia_{1}^{c}\wedge a_{1}^{c}$ is the field strength of the
$SU(N)$ color field $a_{1}^{c}$. $\psi$ and $\tilde{\psi}$ are two left-handed
Weyl fermions in representations ${\cal R}$ and $\overline{\cal R}$ under
$SU(N)$, respectively, constituting together a single Dirac fermion. The
covariant derivative is $D_{\mu}=\partial_{\mu}-ia_{\mu}^{c}$ and $y$ is the
Yukawa coupling. The potential of the complex scalar field is
$V(\Phi)=\lambda\left(|\Phi|^{2}-v^{2}/2\right)$, where $\lambda$ is ${\cal
O}(1)$ parameter. We take $v\gg\Lambda$, where $\Lambda$ is the strong scale
of the gauge sector. The Lagrangian (99) is invariant under two classical
$0$-form symmetries $U(1)_{B}^{(0)}\times U(1)^{(0)}_{\chi}$, the baryon-
number and axial symmetries. The ABJ anomaly in the color background breaks
$U(1)_{\chi}^{(0)}$ down to $\mathbb{Z}^{\chi(0)}_{2T_{\cal R}}$, and we find
that the full good global symmetry of the $SU(N)$ axion-YM theory is
Anber:2019nze ; Anber:2020gig

$\displaystyle
G^{\scriptsize\mbox{global}}=\frac{U(1)^{(0)}_{B}\times\mathbb{Z}^{\chi(0)}_{2T_{\cal
R}}}{\mathbb{Z}_{N/m}\times\mathbb{Z}_{2}^{F}}\times\mathbb{Z}_{m}^{(1)}\,.$
(100)

The $1$-form global symmetry $\mathbb{Z}_{m}^{(1)}$ acts on Wilson’s lines of
$a_{1}^{c}$, where $m=\mbox{gcd}(N,n)$ and $n$ is the $N$-ality of the
representation ${\cal R}$, i.e., the boxes in the Young tableaux modulo $N$.
The baryon-number $U(1)_{B}^{(0)}$ and the chiral
$\mathbb{Z}^{\chi(0)}_{2T_{\cal R}}$ symmetries act on the local fields as

$\displaystyle U(1)_{B}^{(0)}$ $\displaystyle:$
$\displaystyle\quad\psi\rightarrow
e^{i\alpha}\psi\,,\quad\tilde{\psi}\rightarrow
e^{-i\alpha}\tilde{\psi}\,,\quad\Phi\rightarrow\Phi\,,$
$\displaystyle\mathbb{Z}^{\chi(0)}_{2T_{\cal R}}$ $\displaystyle:$
$\displaystyle\quad\psi\rightarrow e^{i\frac{2\pi\ell}{2T_{\cal
R}}}\psi\,,\quad\tilde{\psi}\rightarrow e^{i\frac{2\pi\ell}{2T_{\cal
R}}}\tilde{\psi}\,,\quad\Phi\rightarrow e^{-i\frac{4\pi\ell}{2T_{\cal
R}}}\Phi\,,$ (101)

and $\ell=1,2,..,2T_{\cal R}$ and $T_{\cal R}$ is the Dynkin index of ${\cal
R}$ (in our normalization, $T_{\Box}=1$, where $\Box$ is the fundamental
representation). The modding by $\mathbb{Z}_{N/m}\times\mathbb{Z}_{2}^{F}$ in
(100) is important to remove redundancies. Here, $\mathbb{Z}_{2}^{F}$ is the
$(-1)^{F}$ fermion number subgroup of the Lorentz group. The complex scalar
field can be written as $\Phi=|\Phi|e^{ia}$, where $a$ is the axion. At energy
scales $\ll v$, we can set $|\Phi|=v/\sqrt{2}$, and thus, one may only work
with the axion as the lightest degree of freedom.

###  5.1 The mixed ’t Hooft anomaly and IR Lagrangian

Energy scale $E\gg v$

Among the anomalies of the axion-YM theory, the mixed anomaly between the
$\mathbb{Z}_{m}^{(1)}$ $1$-form center and $\mathbb{Z}^{\chi(0)}_{2T_{\cal
R}}$ chiral symmetries is essential in connection with the $3$-form gauge
theory. To see the link, we first review this anomaly from the UV point of
view Kapustin:2014gua ; Gaiotto:2017yup ; Anber:2020xfk ; Anber:2021lzb . We
shall be general and examine the anomaly between a subgroup of the full center
$\mathbb{Z}_{m}^{(1)}$ and chiral symmetries. We shall also work in the
Euclidean space.

To this end, we turn on a background of
$\mathbb{Z}_{p}^{(1)}\subseteq\mathbb{Z}_{m}^{(1)}$. This can be implemented
by introducing the pair of $U(1)$ fields $(B_{1}^{c},B_{2}^{c})$ and the
constraint $pB_{2}^{c}=dB_{1}^{c}$. Demanding the quantization condition
$\int_{{\cal M}_{2}}dB_{1}^{c}\in 2\pi\mathbb{Z}$ implies the fractional
quantization of $B_{2}^{c}$ flux: $\int_{{\cal
M}_{2}}B_{2}^{c}\in\frac{2\pi\mathbb{Z}}{p}$. We couple $B_{2}^{c}$ to
fermions as follows. First, we enlarge the gauge group from $SU(N)$ to $U(N)$;
we introduce the $\hat{a}_{1}^{c}$ gauge field of $U(N)$ such that
$\hat{a}_{1}^{c}\equiv a_{1}^{c}+\frac{B_{1}^{c}}{p}$ with field strength
$\hat{f}_{2}^{c}=d\hat{a}_{1}^{c}+\hat{a}_{1}^{c}\wedge\hat{a}_{1}^{c}$. This,
in turn, implies the relation $\mbox{tr}(\hat{f}_{2}^{c})=NB_{2}^{c}$.
Enlarging the group form $SU(N)$ to $U(N)$ introduces an extra degree of
freedom, which can be eliminated by postulating the invariance of the theory
under the action of an additional $U(1)^{(1)}$ $1$-form gauge symmetry:
$\hat{a}^{c}_{1}\rightarrow\hat{a}^{c}_{1}-\lambda_{1}^{c}$. This implies that
$\hat{f}_{2}^{c}$, $B_{1}^{c}$, and $B_{2}^{c}$ transform as
$\hat{f}_{2}^{c}\rightarrow\hat{f}_{2}^{c}-d\lambda^{c}_{1}$,
$B_{1}^{c}\rightarrow B_{1}^{c}-p\lambda_{1}^{c}$, and $B_{2}^{c}\rightarrow
B_{2}^{c}-d\lambda_{1}^{c}$, such that the condition $dB_{1}^{c}=pB_{2}^{c}$
remains invariant.

The mixed anomaly between the $\mathbb{Z}_{p}^{(1)}$ center and
$\mathbb{Z}^{\chi(0)}_{2T_{\cal R}}$ chiral symmetries is envisaged by
examining the partition function in the background of the $U(N)$ and
$\mathbb{Z}_{p}^{(1)}$ fluxes. In such backgrounds, the topological charge is
determined by replacing $f_{2}^{c}$ with the combination
$\hat{f}_{2}^{c}-B^{c}_{2}$ in the expression for the topological charge.
Importantly, this expression remains invariant under gauge transformations by
$\lambda_{1}^{c}$. Thus, the topological charge is

$\displaystyle Q^{c}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}\int_{{\cal
M}_{4}}\mbox{tr}_{\Box}\left[\left(\hat{f}_{2}^{c}-B^{c}_{2}\right)\wedge\left(\hat{f}_{2}^{c}-B^{c}_{2}\right)\right]$
(102) $\displaystyle=$ $\displaystyle\frac{1}{8\pi^{2}}\int_{{\cal
M}_{4}}\mbox{tr}_{\Box}\left[\hat{f}_{2}^{c}\wedge\hat{f}_{2}^{c}\right]-\frac{N}{8\pi^{2}}\int_{{\cal
M}_{4}}B_{2}^{c}\wedge B_{2}^{c}\,,$

and is fractional. Recalling that $\int_{{\cal
M}_{4}}\mbox{tr}_{\Box}\left[\hat{f}_{2}^{c}\wedge\hat{f}_{2}^{c}\right]\in
8\pi^{2}\mathbb{Z}$, the partition function transforms by the phase
$-\frac{N}{8\pi^{2}}\int_{{\cal M}_{4}}B_{2}^{c}\wedge
B_{2}^{c}=-\frac{N}{8\pi^{2}p^{2}}\int_{{\cal M}_{4}}dB_{1}^{c}\wedge
dB_{1}^{c}\in\frac{N\mathbb{Z}}{p^{2}}$. This is the mixed anomaly between the
$\mathbb{Z}_{p}^{(1)}$ $1$-form center and $\mathbb{Z}_{2T_{\cal R}}^{(0)}$
discrete chiral symmetries. The anomaly is nontrivial, provided that $p^{2}$
is not a divisor of $N$. It is important to highlight the group-theoretical
result

$\displaystyle\mathbb{Z}_{\scriptsize
m/\mbox{gcd}(m,m^{\prime})}\subseteq\mathbb{Z}_{T_{\cal R}}\,,$ (103)

where we have expressed $N=mm^{\prime}$. This result can be verified
numerically; we shall use it in our analysis below.

Energy scale $\Lambda\ll E\ll v$

At energy scale $\Lambda\ll E\ll v$, the magnitude of $\Phi$ freezes and we
may set $\Phi\cong\frac{v}{\sqrt{2}}e^{ia}$. Also, the fermions acquire a mass
$\sim yv$ and decouple. Then, the effective Lagrangian is:

$\displaystyle{\cal L}_{\Lambda\ll E\ll
v}=-\frac{1}{2g^{2}}\mbox{tr}\left(f_{2}^{c}\wedge\star
f_{2}^{c}\right)+\frac{v^{2}}{2}da\wedge\star da+T_{{\cal R}}aq^{c}\,,$ (104)

where $q^{c}$ the topological charge density: $Q^{c}=\int_{{\cal
M}_{4}}q^{c}$, and $Q^{c}$ is given by the expression (102). Thus, we have

$\displaystyle
q^{c}=\frac{1}{8\pi^{2}}\left[\mbox{tr}_{\Box}\left(\hat{f}_{2}^{c}\wedge\hat{f}_{2}^{c}\right)-NB_{2}^{c}\wedge
B_{2}^{c}\right]\,.$ (105)

In particular, one can easily see that the Euclidean version of (104)
reproduces the anomaly $e^{-i\frac{2\pi N}{p^{2}}}$ under the transformation
$a\rightarrow a+\frac{2\pi}{T_{\cal R}}$.

In the absence of the center background, the Lagrangian (104) is invariant
under the global symmetry group151515In fact, $U(1)^{(2)}$ is only approximate
global symmetry. See our discussion after Eq. (113).

$\displaystyle G^{\scriptsize\mbox{global}}=\mathbb{Z}_{T_{\cal
R}}^{(0)}\times\left(\mathbb{Z}_{N}^{(1)}\tilde{\times}U(1)^{(2)}\right)\,.$
(106)

The $2$-form symmetry $U(1)^{(2)}$ is an emergent winding-number symmetry that
acts on axion strings, while $\mathbb{Z}_{N}^{(1)}$ is an emergent $1$-form
symmetry resulting from the decoupling of fermions. Notice that there can be a
higher group structure between $\mathbb{Z}_{N}^{(1)}$ and $U(1)^{(2)}$
symmetries, and we used the symbol $\tilde{\times}$ to denote this structure.
To see it, we activate a background for $\mathbb{Z}_{N}^{(1)}$ by introducing
the pair $(B_{1}^{(N)},B_{2}^{(N)})$ such that $NB^{(N)}_{2}=dB_{1}^{(N)}$ and
demanding $\int_{M_{2}}dB_{1}^{(N)}\in 2\pi\mathbb{Z}$. This, in turn, implies
the flux of $B_{2}^{(N)}$ is fractional:
$\int_{M_{2}}B_{2}^{(N)}\in\frac{2\pi\mathbb{Z}}{N}$. The pair of fields
$(B_{1}^{(N)},B_{2}^{(N)})$ transforms under a $U(1)^{(1)}$ gauge
transformation as $B_{1}^{(N)}\rightarrow B_{1}^{(N)}+N\lambda_{1}^{(N)}$ and
$B_{2}^{(N)}\rightarrow B_{2}^{(N)}+d\lambda_{1}^{(N)}$, which leaves the
relation $NB^{(N)}_{2}=dB_{1}^{(N)}$ invariant. We also introduce $C_{3}$, the
background gauge field of the global $U(1)^{(2)}$ symmetry.

Inspection of (104, 105) reveals that the backgrounds of
$\mathbb{Z}_{N}^{(1)}$ and $U(1)^{(2)}$ couple to the axion via the term
Choi:2023pdp

$\displaystyle{\cal L}\supset\frac{1}{2\pi}aG_{4}\,,$ (107)

where $G_{4}$ is the field strength of the combined backgrounds. It is given
by

$\displaystyle G_{4}=dC_{3}-\frac{T_{\cal R}N}{4\pi}B_{2}^{(N)}\wedge
B_{2}^{(N)}\,.$ (108)

$G_{4}$ is invariant under a gauge transformation by $\lambda_{1}^{(N)}$
provided that $C_{3}$ transforms as

$\displaystyle C_{3}\rightarrow C_{3}+d\lambda_{2}+\frac{T_{\cal
R}N}{2\pi}d\lambda_{1}^{(N)}\wedge B_{2}^{(N)}+\frac{T_{\cal
R}N}{4\pi}d\lambda_{1}^{(N)}\wedge d\lambda_{1}^{(N)}\,.$ (109)

The interplay among $C_{3}$, $B_{2}^{(N)}$, and $\lambda_{1}^{(N)}$ indicates
a higher-group structure, where $\mathbb{Z}_{N}^{(1)}$ represents the daughter
symmetry and $U(1)^{(2)}$ the parent symmetry. Notably, the former cannot
exist independently of the latter Cordova:2018cvg , imposing constraints on
the emergent symmetry scales: $E_{\mathbb{Z}_{N}^{(1)}}\lesssim
E_{U(1)^{(2)}}$. This condition aligns well with effective field theory
expectations: $E_{\mathbb{Z}_{N}^{(1)}}\cong\sqrt{\lambda}v$,
$E_{\mathbb{Z}_{N}^{(1)}}\cong yv$, and $\lambda\ll y^{2}\ll 1$; see
Brennan:2020ehu  for details.

In a higher-group structure, one cannot gauge the daughter symmetry without
gauging the parent. But the reverse is possible. This observation shall play
an important role below. Notice that the higher-group symmetry becomes split
(trivialized) if one can write $G_{4}$ as a total derivative Choi:2023pdp .
For example, there is no higher-group structure between the genuine
$\mathbb{Z}_{m}^{(1)}$ symmetry of the UV theory and $U(1)^{(2)}$. To see
that, we replace $B_{2}^{(N)}\wedge B_{2}^{(N)}$ in Eq. (108) by
$B_{2}^{c}\wedge B_{2}^{c}$, where we use the pair $(B_{1}^{c},B_{2}^{c})$
(which satisfies the constraint $mB_{2}^{c}=dB_{1}^{c}$) to activate the
background of $\mathbb{Z}_{m}^{(1)}$. Thus, $G_{4}=dC_{3}-\frac{T_{\cal
R}N}{4\pi m^{2}}dB_{1}^{c}\wedge dB_{1}^{c}$. Since $\mathbb{Z}_{\scriptsize
m/\mbox{gcd}(m,m^{\prime})}\subseteq\mathbb{Z}_{T_{\cal R}}$ (remember that
$N=mm^{\prime}$), we can write $\frac{T_{\cal R}N}{m^{2}}=m^{\prime\prime}$,
$m^{\prime\prime}\in\mathbb{N}$, and hence, $G_{4}=dC_{3}^{\prime}\equiv
d\left(C_{3}-\frac{m^{\prime\prime}}{4\pi}B_{1}^{c}\wedge dB_{1}^{c}\right)$,
trivializing the higher-group. This observation is important for our
subsequent analysis.

Energy scale $E\ll\Lambda$

Next, we flow to the deep IR at energy scale $\ll\Lambda$, where we assume the
theory confines, and hence, the color gauge field is gapped. We must write
down an effective Lagrangian that captures the UV center-chiral anomaly. This
can be achieved by (i) gauging the $U(1)^{(2)}$ symmetry and introducing the
dynamical $3$-form gauge field $c_{3}$ and (ii) replacing
$\mbox{tr}_{\Box}\left(\hat{f}_{2}^{c}\wedge\hat{f}_{2}^{c}\right)/4\pi$ in
Eqs. (104, 105) by $dc_{3}$:

$\displaystyle{\cal L}_{E\ll\Lambda}=\frac{v^{2}}{2}da\wedge\star
da+\frac{T_{{\cal R}}a}{2\pi}\left(dc_{3}-\frac{N}{4\pi}B_{2}^{c}\wedge
B_{2}^{c}\right)+\Lambda^{4}{\cal
K}\left(\frac{dc_{3}-\frac{N}{4\pi}B_{2}^{c}\wedge
B_{2}^{c}}{\Lambda^{4}}\right)\,,$

and we added a kinetic energy term for $c_{3}$. The field strength of $c_{3}$
satisfies the quantization condition $\int_{{\cal M}_{4}}dc_{3}\in
2\pi\mathbb{Z}$, which is simply the infrared manifestation of the
quantization of topological charges in Yang-Mills theory. The reader will
notice that the coupling between $a$ and $dc_{3}$ has an extra factor of
$T_{\cal R}$ compared to the coupling in Eq. (107). This is because $c_{3}$ in
Eq. (5.1) is a dynamical rather than a background field, and as the dynamical
field absorbs the axion, it should describe the formation of $T_{\cal R}$
domain walls. As noted above, we are allowed to gauge $U(1)^{(2)}$ without
gauging the daughter symmetry $\mathbb{Z}_{N}^{(1)}$. This is important;
otherwise, we would have changed the theory’s global structure and run into
trouble since $\mathbb{Z}_{N}^{(1)}$ is not a genuine symmetry of the theory.
The Lagrangian (5.1) must pass several checks. First, it must be invariant
under the $\mathbb{Z}_{2T_{\cal R}}^{\chi(0)}$ chiral symmetry in the absence
of the center background, which is evident from the transformation
$a\rightarrow a+\frac{2\pi}{T_{\cal R}}$ along with the condition $\int_{{\cal
M}_{4}}dc_{3}\in 2\pi\mathbb{Z}$. Second, the Lagrangian must be invariant
under the same auxiliary $U(1)^{(1)}$ gauge transformation, by
$\lambda_{1}^{c}$, of the UV theory. This is the case provided that $c_{3}$
transforms as

$\displaystyle c_{3}\rightarrow
c_{3}+d\lambda_{2}+\frac{p^{\prime}}{2\pi}B_{1}^{c}\wedge
d\lambda_{1}^{c}+\frac{pp^{\prime}}{4\pi}\lambda_{1}^{c}\wedge
d\lambda_{1}^{c}\,,$ (111)

and we wrote $N=pp^{\prime}$. Also, the Lagrangian (5.1) must reproduce the
mixed $\mathbb{Z}_{2T_{\cal R}}^{\chi(0)}$-$\mathbb{Z}_{p}^{(1)}$ anomaly of
the UV theory. This can be easily verified by observing that the partition
function acquires the phase $e^{-i\frac{2\pi N}{p^{2}}}$ when $a$ is shifted
by $a\rightarrow a+\frac{2\pi}{T_{\cal R}}$ in the presence of the center
background. In the absence of a center background, the Lagrangian (5.1)
exactly matches (22) in Section 2.1, and everything we said there applies
here.

Another check161616We thank T. Tanizaki for pointing this out. on the validity
of (5.1) is to integrate out $c_{3}$ along the lines of our discussion that
led from Eq. (31) to Eq. (34). Thus, we sum over arbitrary values of the
integers $\int_{{\cal M}_{4}}f_{4}\in 2\pi\mathbb{Z}$ and use the Poisson
resummation formula $\sum_{m\in\mathbb{Z}}\delta\left(2\pi m-\int_{{\cal
M}_{4}}f_{4}\right)=\sum_{k\in\mathbb{Z}}e^{-ik\int_{{\cal M}_{4}}f_{4}}$. We
also use the change of variables
$\hat{f}_{4}=f_{4}-\frac{N}{4\pi}B_{2}^{c}\wedge B_{2}^{c}$. Specifying to the
canonical kinetic term ${\cal K}_{\scriptsize\mbox{can}}$, performing the
Gaussian integral over $\hat{f}_{4}$, and focusing only on the zero modes, we
obtain the Euclidean partition function

$\displaystyle Z[a]\sim\sum_{k\in\mathbb{Z}}e^{-i\frac{kN}{4\pi}\int_{{\cal
M}_{4}}B_{2}^{c}\wedge B_{2}^{c}}e^{-\frac{\Lambda^{4}V_{{\cal
M}_{4}}}{8\pi^{2}}\left(T_{\cal R}a+2\pi k\right)^{2}}\,.$ (112)

This effective partition function picks up the anomaly
$e^{i\frac{N}{4\pi}\int_{{\cal M}_{4}}B_{2}^{c}\wedge
B_{2}^{c}}=e^{-i\frac{2\pi N}{p^{2}}}$ upon shifting $a\rightarrow
a+\frac{2\pi}{T_{\cal R}}$. It also displays the expected structure of the
Yang-Mills theory: it has an infinite number of vacua, with the true vacuum
energy density given by

$\displaystyle V(a)=\frac{\Lambda^{4}}{8\pi^{2}}\mbox{min}_{k}\left(T_{\cal
R}a+2\pi k\right)^{2}\,.$ (113)

The potential $V(a)$ has $T_{\cal R}$ minima with cusps at $a=\pi/T_{\cal
R},3\pi/T_{\cal R}$, etc.. The cusps indicate that the potential $V(a)$ is
missing degrees of freedom at these locations. These are the hadronic walls
sandwiched between the axion domain walls. An axion wall has width $\sim
v/\Lambda^{2}$, while a hadronic wall is much thinner with width
$\sim\Lambda^{-1}$. Including the infinite sum over all the integers
$\int_{{\cal M}_{4}}f_{4}\in 2\pi m$, $m\in\mathbb{Z}$ was crucial to see
these cusps. As emphasized above, the integer $m$ is the IR manifestation of
the Yang-Mills instantons’ topological charge. Below $\Lambda$, the theory is
strongly coupled, and the vacuum receives contributions from all the
topological charge sectors.

Let us examine the theory’s behavior at energy scales $\Lambda\ll E\ll v$,
such as at a corresponding temperature. In this case, it suffices to include
the contribution from minimal charges $\int_{{\cal M}_{4}}f_{4}=m=\pm 1$ in
the partition function. Translating this into the language of Yang-Mills
instantons, the dilute instanton-gas approximation is reliable at this
temperature because it serves as an infrared cut-off on the instanton’s scale
modulus Gross:1980br . Thus, summing over the smallest instantons, which
possess topological charges of $\pm 1$, is adequate. Limiting the sum over $m$
to the lowest charge sector means that one can no longer perform the Poisson
resummation that leads to (112), and thus, one can no longer make sense of
(113) or the cusps. This is consistent with the expectation that the hadronic
walls melt away at a temperature $\sim\Lambda$. Nevertheless, at temperatures
in the range $\Lambda\ll T\ll v$, the classical theory (5.1) still possesses
$T_{\cal R}$ vacua with axion domain walls interpolating between them. Notice
that in this energy range, $c_{3}$ does not strongly fluctuate
(since$\int_{{\cal M}_{4}}f_{4}=\pm 1$), and we can consider the $U(1)^{(2)}$
$3$-form gauge field $c_{3}$ as a background rather than a dynamical field.
Thus, one may still regard $U(1)^{(2)}$ as an approximate global symmetry.
Eventually, the axion domain walls melt at temperature $T\gtrsim v$.

As noted in Luscher:1978rn ; Gabadadze:2002ff ; Dvali:2005an  and discussed in
the Introduction, it was recognized that the IR behavior of both pure YM
theory in the large-$N$ limit and the axion-YM theory can be effectively
described using the $3$-form gauge field $c_{3}$. Here, incorporating $c_{3}$
in our discussion has been essential for aligning with the infrared
constraints of the ’t Hooft anomaly. Our method provides a systematic approach
to argue for a consistent infrared effective field theory of the axion-YM
system.

At energy scales below $\Lambda$, the theory acquires the global symmetry

$\displaystyle
G^{\scriptsize\mbox{global}}=U(1)^{(-1)}\times\mathbb{Z}_{T_{\cal
R}}^{(0)}\times\left(\mathbb{Z}_{N}^{(1)}\tilde{\times}\mathbb{Z}_{T_{\cal
R}}^{(3)}\right)\,.$ (114)

In general, a higher-group structure may exist between $\mathbb{Z}_{N}^{(1)}$
and $\mathbb{Z}_{T_{\cal R}}^{(3)}$, which becomes apparent when both
symmetries’ backgrounds are activated. The background of
$\mathbb{Z}_{N}^{(1)}$ was discussed earlier, while that of
$\mathbb{Z}_{T_{\cal R}}^{(3)}$ can be activated by introducing the pair
$(F_{3},F_{4})$, satisfying the constraint $dF_{3}=T_{\cal R}F_{4}$, along
with the quantization condition $\int_{{\cal M}4}dF_{3}\in 2\pi\mathbb{Z}$
Tanizaki:2019rbk . The axion coupled to these backgrounds is represented by

$\displaystyle{\cal L}\supset\frac{a}{2\pi}\left(T_{\cal
R}dc_{3}+dF_{3}-\frac{T_{\cal R}N}{4\pi}B_{2}^{(N)}\wedge B_{2}^{(N)}\right).$
(115)

Maintaining invariance under a gauge transformation by $\lambda_{1}^{(N)}$
requires $F_{3}$ to transform as

$\displaystyle F_{3}\rightarrow F_{3}+d\lambda_{2}+\frac{T_{\cal
R}N}{2\pi}d\lambda_{1}^{(N)}\wedge B_{2}^{(N)}+\frac{T_{\cal
R}N}{4\pi}d\lambda_{1}^{(N)}\wedge d\lambda_{1}^{(N)}\,.$ (116)

The interplay among $F_{3}$, $B_{2}^{(N)}$, and $\lambda_{1}^{(N)}$ indicates
a higher-group symmetry $\mathbb{Z}_{N}^{(1)}\tilde{\times}\mathbb{Z}_{T_{\cal
R}}^{(3)}$. However, this higher-group structure becomes trivial if
$dF_{3}-\frac{NT_{\cal R}}{4\pi N^{2}}dB_{1}^{(N)}\wedge dB_{1}^{(N)}$ can be
expressed as a total derivative. This holds particularly true for the
$\mathbb{Z}_{m}^{(1)}$ symmetry, as demonstrated above in a similar case
involving $U(1)^{(2)}$ and $\mathbb{Z}_{m}^{(1)}$. Understanding this aspect
is pivotal when gauging the genuine center, as this operation should be
executed without gauging $\mathbb{Z}_{T_{\cal R}}^{(3)}$.

In the IR, the three symmetries $U(1)^{(-1)}$, $\mathbb{Z}_{T_{\cal
R}}^{(0)}$, and $\mathbb{Z}_{T_{\cal R}}^{(3)}$ undergo spontaneous breaking.
The emergent symmetry $\mathbb{Z}_{N}^{(1)}$ remains unbroken until length
scales $\sim yv/\Lambda^{2}$, at which point it also undergoes explicit
breaking due to the heavy fermions that pop up from vacuum as we take the
Wilson lines to be larger than $\sim yv/\Lambda^{2}$. This leaves
$\mathbb{Z}_{m}^{(1)}$ as the sole surviving unbroken symmetry.

###  5.2 $SU(N)/\mathbb{Z}_{p}$ and noninvertible chiral symmetry

Let us investigate whether our construction yields the desired results when we
gauge the genuine center or any of its subgroups, aligning with the well-
established findings in the literature Kaidi:2021xfk ; Argurio:2023lwl ;
Anber:2023mlc . We shall see that the answer is affirmative, lending support
to the picture that the deep IR regime of the system is genuinely described by
the $3$-form gauge theory.

We consider the same axion-YM theory with matter, but now let us gauge a
subgroup of the center $\mathbb{Z}_{p}^{(1)}\subseteq\mathbb{Z}_{m}^{(1)}$,
i.e., we consider $SU(N)/\mathbb{Z}_{p}$ axion-YM theory with matter171717In
principle, there are $p$ distinct theories: $(SU(N)/\mathbb{Z}_{p})_{k}$,
$k=0,1,..,p-1$ differing by the admissible genuine (electric, magnetic, or
dyonic) line operators Aharony:2013hda . In this paper, we limit our treatment
to $(SU(N)/\mathbb{Z}_{p})_{k=0}$. The Hilbert space and the noninvertible
chiral symmetry in $(SU(N)/\mathbb{Z}_{p})_{k=0}$ theory were considered in
Anber:2023mlc .. This theory is constructed by promoting
$(B_{1}^{c},B_{2}^{c})$ to dynamical fields $(b_{1}^{c},b_{2}^{c})$ and
performing the sum over the fractional instantons in the path integral. Let us
define the new $3$-form gauge field $\hat{c}_{3}$:

$\displaystyle\hat{c}_{3}\equiv c_{3}-\frac{N}{4\pi p^{2}}b_{1}^{c}\wedge
db_{1}^{c}\,,$ (117)

keeping in mind the quantization condition $\int_{{\cal M}_{4}}dc_{3}\in
2\pi\mathbb{Z}$. The Lagrangian of this theory at energy scale $E\ll\Lambda$
reads

$\displaystyle{\cal
L}_{E\ll\Lambda}[(b_{1}^{c},b_{2}^{c})]=\frac{v^{2}}{2}da\wedge\star
da+\frac{T_{\cal R}}{2\pi}a\wedge d\hat{c}_{3}+\Lambda^{4}{\cal
K}\left(\frac{d\hat{c}_{3}}{\Lambda^{4}}\right)\,,$ (118)

and the partition function is

$\displaystyle Z=\sum_{(b_{1}^{c},b_{2}^{c})}\int[Dc_{3}][Da]e^{i\int_{{\cal
M}_{4}}{\cal L}_{E\ll\Lambda}[(b_{1}^{c},b_{2}^{c})]}\,.$ (119)

In the Kalb-Ramond frame, we replace $q\rightarrow T_{\cal R}$ and
$c_{3}\rightarrow\hat{c}_{3}$ in (84). It is important to repeat what we
stated above: we can gauge the genuine $\mathbb{Z}_{m}^{(1)}$ center symmetry
or a subgroup thereof without spoiling $\mathbb{Z}_{T_{\cal R}}^{(3)}$ since
the pair does not constitute a higher-group. On the contrary, gauging the
emergent $\mathbb{Z}_{N}^{(1)}$ is disastrous: it entails that we also gauge
$\mathbb{Z}_{T_{\cal R}}^{(3)}$, which destroys the domain walls.

The chiral symmetry defect is given by (24) after replacing $c_{3}$ with
$\hat{c}_{3}$ and summing over $(b_{1}^{c},b_{2}^{c})$:

$\displaystyle\tilde{U}_{\ell}^{(0)}(\mathcal{M}_{3})\sim\sum_{(b_{1}^{c},b_{2}^{c})}e^{i\frac{2\pi\ell}{T_{\cal
R}}\int_{\mathcal{M}_{3}}v^{2}\star da-i\ell\int_{{\cal
M}_{3}}\left(c_{3}-\frac{N}{4\pi p^{2}}b_{1}^{c}\wedge
db_{1}^{c}\right)}\,,\quad\ell=1,2,..,T_{{\cal R}}\,.$ (120)

The new defect $\tilde{U}_{\ell}^{(0)}(\mathcal{M}_{3})$ defines a
noninvertible chiral symmetry $\tilde{\mathbb{Z}}_{T_{\cal R}}^{(0)}$. To
simplify the form of $\tilde{U}_{\ell}^{(0)}(\mathcal{M}_{3})$, we write $N$
as $N=pp^{\prime}$ and assume that $p^{\prime}=1$ Mod $p$. Then,
$\frac{N}{4\pi p^{2}}b_{1}^{c}\wedge db_{1}^{c}\sim\frac{1}{4\pi
p}b_{1}^{c}\wedge db_{1}^{c}$, i.e., this is an improperly quantized quantum
Hall term. We may rewrite it in terms of an auxiliary $1$-form gauge field
$\varphi_{1}$ that lives on ${\cal M}_{3}$:

$\displaystyle\tilde{U}_{\ell}^{(0)}(\mathcal{M}_{3})\sim\sum_{(b_{1}^{c},b_{2}^{c})}e^{i\frac{2\pi\ell}{T_{\cal
R}}\int_{\mathcal{M}_{3}}v^{2}\star da-i\ell\int_{{\cal
M}_{3}}c_{3}}\int[D\varphi_{1}]e^{-i\ell\int_{{\cal
M}_{3}}\left(\frac{p}{4\pi}\varphi_{1}\wedge
d\varphi_{1}+\frac{1}{2\pi}\varphi_{1}\wedge db_{1}^{c}\right)}\,.$ (121)

The last term is the minimal abelian TQFT ${\cal A}^{p,1}$ discussed in
Hsin:2018vcg . When $\ell$ is a multiple of $p$, the TQFT is trivialized,
giving us an invertible symmetry.

##  6 Discussion

In this letter, we critically evaluated the hypothesis that the $3$-form gauge
theory offers more than an alternative framework for the deep IR regime of
axion-Yang-Mills systems. We commenced by rigorously investigating the
$3$-form gauge theory coupled to axions, as encapsulated by the Lagrangian
(22). Our analysis revealed that this theory represents a network of domain
walls terminating on an axion string, with particular emphasis placed on its
global structure. A dual formulation, the Kalb-Ramond Lagrangian (84), was
also considered, which describes the same physical phenomenon. Notably, while
some symmetry defects are explicit in one formulation, others are evident in
the alternative frame. Crucially, in the absence of gravitational
effects181818Inclusion of gravity may differentiate between the axion and
Kalb-Ramond frames, see Duff:1980qv ., both formulations are equivalent,
possessing identical global symmetries.

Subsequently, we examined the $SU(N)$ Yang-Mills theory with a Dirac fermion
coupled to an $SU(N)$-neutral complex scalar, highlighting the necessity of an
emergent $3$-form gauge field for IR matching of the theory’s mixed center-
chiral anomaly. Consider varying the complex scalar vev such that we go to the
limit $v\ll\Lambda$. In this case, the strong dynamics set in well before the
axions are amenable to the weak-coupling treatment. In this opposite limit,
the theory still forms domain walls leading to $T_{\cal R}$ distinct vacua,
and thus, we do not expect a bulk phase transition to take place as we vary
$v$ below or above $\Lambda$. We may not rigorously justify the introduction
of the $3$-form gauge field in this scenario. However, by continuity, we
expect that the $3$-form gauge theory remains a valid description in the deep
IR. This reasoning, combined with the large-$N$ limit analysis discussed in
the introduction, supports the notion that the vacuum of Yang-Mills theory
should likely be described by a $3$-form gauge theory.

Incorporating gravity offers further insights into the significance of the
$3$-form description, as the cosmological constant in this context can be
interpreted as arising from a gauge principle. Theoretically, one can
distinguish between a pure cosmological constant and a $3$-form gauge theory,
as the latter yields a non-vanishing contribution to the trace anomaly
proportional to the Gauss-Bonnet invariant Duff:1980qv ; Christensen:1978md .
Irrespective of this subtle effect, Brown and Teitelboim realized that the
action (2) taken as a starting point with no reference to its UV completion
leads to the quantum creation of closed membranes localized on the boundary of
${\cal M}_{4}$ Brown:1987dd ; Brown:1988kg . As the membranes are produced,
the vacuum energy density associated with $c_{3}$ decreases, reducing the
effective value of the cosmological constant. This idea, when refined, may
lead to a solution to the cosmological constant problem Bousso:2000xa ;
Bousso:2007gp . Also, connections between the QCD vacuum and the cosmological
constant problem were discussed in Kaloper:2017fsa ; Urban:2009vy .

As discussed in the paper, introducing the $3$-form gauge field $c_{3}$ was
necessary to match the chiral-center anomaly in an axion-Yang-Mills system.
Eventually, $c_{3}$ eats the axion, becoming a short-range field, and the
cosmological constant vanishes. Yet, one can think of an alternative scenario
with an axion, two distinct Yang-Mills fields, and a chiral-center anomaly. In
this case, two $3$-form gauge fields are anticipated. One combination of these
fields eats the axion, while an orthogonal combination remains gapless. The
latter can source a cosmological constant. Intriguingly, in this scenario, the
infrared cosmological constant can be considered a by-product of the ’t Hooft
anomaly-matching condition. However, in such a scenario, global symmetries
should only be considered approximate since exact global symmetries are
forbidden in quantum gravity Harlow:2018tng .

The method presented in this paper can be extended in various ways. One
immediate application is to address the problem of multi-flavor quarks by
incorporating the $3$-form gauge theory description in the chiral Lagrangian.
Another venue is applying our formalism to the Standard Model (SM) and its
variants, potentially through coupling with axions. It is well-established
that the SM exhibits a $\mathbb{Z}_{6}^{(1)}$ $1$-form symmetry, and the true
gauge group might be modded by $\mathbb{Z}_{6}$ or a subgroup thereof (see,
e.g., Tong:2017oea ; Anber:2021upc ; Wang:2018jkc ; Wan:2019gqr ; Li:2024nuo ;
Alonso:2024pmq ). Exploring whether and how a $3$-form gauge theory may emerge
deep in the IR of the SM and its extensions and whether the modded discrete
group could play a significant role in this formalism will be an exciting
avenue of research. Additionally, linking the emergent $3$-form to the
observed cosmological constant presents another intriguing possibility. These
investigations are worthy of future exploration.

Acknowledgments:

We thank M. Bullimore, A. Grigoletto, N. Iqbal, T. Jacobson, E. Poppitz, T.
Sulejmanpasic, Y. Tanizaki, M. Ünsal, and A. Zhitnitsky for many discussions.
We also thank E. Poppitz for his comments on the Manuscript. This work is
supported by STFC through grant ST/T000708/1.

This work is dedicated to the memory of my late father, Mostafa, whose love,
support, and encouragement have been a constant source of inspiration. M.A.

## References

  * (1) M. Luscher, The Secret Long Range Force in Quantum Field Theories With Instantons, Phys. Lett. B 78 (1978) 465–467.
  * (2) J. B. Kogut and L. Susskind, How to Solve the eta –$>$ 3 pi Problem by Seizing the Vacuum, Phys. Rev. D 11 (1975) 3594.
  * (3) G. Veneziano, U(1) Without Instantons, Nucl. Phys. B 159 (1979) 213–224.
  * (4) P. Di Vecchia and G. Veneziano, Chiral Dynamics in the Large n Limit, Nucl. Phys. B 171 (1980) 253–272.
  * (5) G. Gabadadze and M. Shifman, QCD vacuum and axions: What’s happening?, Int. J. Mod. Phys. A 17 (2002) 3689–3728, [hep-ph/0206123].
  * (6) E. Witten, Large N Chiral Dynamics, Annals Phys. 128 (1980) 363.
  * (7) A. Aurilia, H. Nicolai, and P. K. Townsend, Hidden Constants: The Theta Parameter of QCD and the Cosmological Constant of N=8 Supergravity, Nucl. Phys. B 176 (1980) 509–522.
  * (8) M. J. Duff and P. van Nieuwenhuizen, Quantum Inequivalence of Different Field Representations, Phys. Lett. B 94 (1980) 179–182.
  * (9) P. G. O. Freund and M. A. Rubin, Dynamics of Dimensional Reduction, Phys. Lett. B 97 (1980) 233–235.
  * (10) M. Nitta and R. Yokokura, Higher derivative three-form gauge theories and their supersymmetric extension, JHEP 10 (2018) 146, [arXiv:1809.03957].
  * (11) M. Nitta and R. Yokokura, Topological couplings in higher derivative extensions of supersymmetric three-form gauge theories, JHEP 05 (2019) 102, [arXiv:1810.12678].
  * (12) R. D. Peccei and H. R. Quinn, Constraints Imposed by CP Conservation in the Presence of Instantons, Phys. Rev. D 16 (1977) 1791–1797.
  * (13) G. Dvali, Three-form gauging of axion symmetries and gravity, hep-th/0507215.
  * (14) G. Dvali, Strong-$CP$ with and without gravity, arXiv:2209.14219.
  * (15) D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172, [arXiv:1412.5148].
  * (16) T. D. Brennan and S. Hong, Introduction to Generalized Global Symmetries in QFT and Particle Physics, arXiv:2306.00912.
  * (17) Y. Hidaka, M. Nitta, and R. Yokokura, Emergent discrete 3-form symmetry and domain walls, Phys. Lett. B 803 (2020) 135290, [arXiv:1912.02782].
  * (18) T. D. Brennan and C. Cordova, Axions, higher-groups, and emergent symmetry, JHEP 02 (2022) 145, [arXiv:2011.09600].
  * (19) R. Yokokura, Non-invertible symmetries in axion electrodynamics, arXiv:2212.05001.
  * (20) N. Yamamoto and R. Yokokura, Topological mass generation in gapless systems, Phys. Rev. D 104 (2021), no. 2 025010, [arXiv:2009.07621].
  * (21) Y. Hidaka, M. Nitta, and R. Yokokura, Global 4-group symmetry and ’t Hooft anomalies in topological axion electrodynamics, PTEP 2022 (2022), no. 4 04A109, [arXiv:2108.12564].
  * (22) J. Kaidi, K. Ohmori, and Y. Zheng, Kramers-Wannier-like Duality Defects in (3+1)D Gauge Theories, Phys. Rev. Lett. 128 (2022), no. 11 111601, [arXiv:2111.01141].
  * (23) C. Cordova and K. Ohmori, Noninvertible Chiral Symmetry and Exponential Hierarchies, Phys. Rev. X 13 (2023), no. 1 011034, [arXiv:2205.06243].
  * (24) 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].
  * (25) L. Bhardwaj, S. Schafer-Nameki, and J. Wu, Universal Non-Invertible Symmetries, Fortsch. Phys. 70 (2022), no. 11 2200143, [arXiv:2208.05973].
  * (26) T. Bartsch, M. Bullimore, A. E. V. Ferrari, and J. Pearson, Non-invertible Symmetries and Higher Representation Theory I, arXiv:2208.05993.
  * (27) M. M. Anber and E. Poppitz, Noninvertible anomalies in SU(N) × U(1) gauge theories, JHEP 08 (2023) 149, [arXiv:2305.14425].
  * (28) S.-H. Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries, arXiv:2308.00747.
  * (29) S. Schafer-Nameki, ICTP lectures on (non-)invertible generalized symmetries, Phys. Rept. 1063 (2024) 1–55, [arXiv:2305.18296].
  * (30) A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001, [arXiv:1401.0740].
  * (31) D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, Theta, Time Reversal, and Temperature, JHEP 05 (2017) 091, [arXiv:1703.00501].
  * (32) N. Seiberg, Y. Tachikawa, and K. Yonekura, Anomalies of Duality Groups and Extended Conformal Manifolds, PTEP 2018 (2018), no. 7 073B04, [arXiv:1803.07366].
  * (33) M. M. Anber and E. Poppitz, Deconfinement on axion domain walls, JHEP 03 (2020) 124, [arXiv:2001.03631].
  * (34) T. Fugleberg, I. E. Halperin, and A. Zhitnitsky, Domain walls and theta dependence in QCD with an effective Lagrangian approach, Phys. Rev. D 59 (1999) 074023, [hep-ph/9808469].
  * (35) G. Gabadadze and M. A. Shifman, D walls and junctions in supersymmetric gluodynamics in the large N limit suggest the existence of heavy hadrons, Phys. Rev. D 61 (2000) 075014, [hep-th/9910050].
  * (36) D. Aloni, E. García-Valdecasas, M. Reece, and M. Suzuki, Spontaneously Broken $(-1)$-Form U(1) Symmetries, arXiv:2402.00117.
  * (37) M. Kalb and P. Ramond, Classical direct interstring action, Phys. Rev. D 9 (1974) 2273–2284.
  * (38) M. M. Anber and E. Poppitz, On the baryon-color-flavor (BCF) anomaly in vector-like theories, JHEP 11 (2019) 063, [arXiv:1909.09027].
  * (39) M. M. Anber and E. Poppitz, Generalized ’t Hooft anomalies on non-spin manifolds, JHEP 04 (2020) 097, [arXiv:2002.02037].
  * (40) M. M. Anber, Condensates and anomaly cascade in vector-like theories, JHEP 03 (2021) 191, [arXiv:2101.04132].
  * (41) Y. Choi, M. Forslund, H. T. Lam, and S.-H. Shao, Quantization of Axion-Gauge Couplings and Noninvertible Higher Symmetries, Phys. Rev. Lett. 132 (2024), no. 12 121601, [arXiv:2309.03937].
  * (42) C. Córdova, T. T. Dumitrescu, and K. Intriligator, Exploring 2-Group Global Symmetries, JHEP 02 (2019) 184, [arXiv:1802.04790].
  * (43) D. J. Gross, R. D. Pisarski, and L. G. Yaffe, QCD and Instantons at Finite Temperature, Rev. Mod. Phys. 53 (1981) 43.
  * (44) Y. Tanizaki and M. Ünsal, Modified instanton sum in QCD and higher-groups, JHEP 03 (2020) 123, [arXiv:1912.01033].
  * (45) R. Argurio and R. Vandepopeliere, When $\mathbb{Z}$2 one-form symmetry leads to non-invertible axial symmetries, JHEP 08 (2023) 205, [arXiv:2306.01414].
  * (46) M. M. Anber and S. Y. L. Chan, Noninvertible symmetries and anomalies from gauging 1-form electric centers, JHEP 03 (2024) 169, [arXiv:2311.07662].
  * (47) O. Aharony, N. Seiberg, and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115, [arXiv:1305.0318].
  * (48) P.-S. Hsin, H. T. Lam, and N. Seiberg, Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d, SciPost Phys. 6 (2019), no. 3 039, [arXiv:1812.04716].
  * (49) S. M. Christensen and M. J. Duff, New Gravitational Index Theorems and Supertheorems, Nucl. Phys. B 154 (1979) 301–342.
  * (50) J. D. Brown and C. Teitelboim, Dynamical Neutralization of the Cosmological Constant, Phys. Lett. B 195 (1987) 177–182.
  * (51) J. D. Brown and C. Teitelboim, Neutralization of the Cosmological Constant by Membrane Creation, Nucl. Phys. B 297 (1988) 787–836.
  * (52) R. Bousso and J. Polchinski, Quantization of four form fluxes and dynamical neutralization of the cosmological constant, JHEP 06 (2000) 006, [hep-th/0004134].
  * (53) R. Bousso, TASI Lectures on the Cosmological Constant, Gen. Rel. Grav. 40 (2008) 607–637, [arXiv:0708.4231].
  * (54) N. Kaloper and J. Terning, Landscaping the Strong CP Problem, JHEP 03 (2019) 032, [arXiv:1710.01740].
  * (55) F. R. Urban and A. R. Zhitnitsky, The cosmological constant from the QCD Veneziano ghost, Phys. Lett. B 688 (2010) 9–12, [arXiv:0906.2162].
  * (56) D. Harlow and H. Ooguri, Symmetries in quantum field theory and quantum gravity, Commun. Math. Phys. 383 (2021), no. 3 1669–1804, [arXiv:1810.05338].
  * (57) D. Tong, Line Operators in the Standard Model, JHEP 07 (2017) 104, [arXiv:1705.01853].
  * (58) M. M. Anber and E. Poppitz, Nonperturbative effects in the Standard Model with gauged 1-form symmetry, JHEP 12 (2021) 055, [arXiv:2110.02981].
  * (59) J. Wang and X.-G. Wen, Nonperturbative definition of the standard models, Phys. Rev. Res. 2 (2020), no. 2 023356, [arXiv:1809.11171].
  * (60) Z. Wan and J. Wang, Beyond Standard Models and Grand Unifications: Anomalies, Topological Terms, and Dynamical Constraints via Cobordisms, JHEP 07 (2020) 062, [arXiv:1910.14668].
  * (61) H.-L. Li and L.-X. Xu, The Standard Model Gauge Group, SMEFT, and Generalized Symmetries, arXiv:2404.04229.
  * (62) R. Alonso, D. Dimakou, and M. West, Fractional-charge hadrons and leptons to tell the Standard Model group apart, arXiv:2404.03438.