11institutetext: Department of Physics, North Carolina State University,
Raleigh, NC 27607, USA22institutetext: Yukawa Institute for Theoretical
Physics, Kyoto University, Kyoto 606-8502, Japan

# Semi-Abelian gauge theories, non-invertible symmetries, and string tensions
beyond $N$-ality

Mendel Nguyen 2 Yuya Tanizaki 1 Mithat Ünsal<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study a $3$d lattice gauge theory with gauge group
$\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}$, which is obtained by gauging the
$\mathrm{S}_{N}$ global symmetry of a pure $\mathrm{U}(1)^{N-1}$ gauge theory,
and we call it the semi-Abelian gauge theory. We compute mass gaps and string
tensions for both theories using the monopole-gas description. We find that
the effective potential receives equal contributions at leading order from
monopoles associated with the entire $\mathrm{SU}(N)$ root system. Even though
the center symmetry of the semi-Abelian gauge theory is given by
$\mathbb{Z}_{N}$, we observe that the string tensions do not obey the
$N$-ality rule and carry more detailed information on the representations of
the gauge group. We find that this refinement is due to the presence of non-
invertible topological lines as a remnant of $\mathrm{U}(1)^{N-1}$ one-form
symmetry in the original Abelian lattice theory. Upon adding charged particles
corresponding to $W$-bosons, such non-invertible symmetries are explicitly
broken so that the $N$-ality rule should emerge in the deep infrared regime.

††preprint: YITP-21-01

##  1 Introduction

In all calculable, confining $\mathrm{SU}(N)$ gauge theories in continuum,
such as the Polyakov model on $\mathbb{R}^{3}$ Polyakov:1975rs ;
Polyakov:1987ez , the Seiberg–Witten model on $\mathbb{R}^{4}$ Seiberg:1994rs
, and deformed Yang–Mills and adjoint QCD on $\mathbb{R}^{3}\times S^{1}$
Unsal:2008ch ; Unsal:2007jx , the gauge dynamics Abelianize to
$\mathrm{U}(1)^{N-1}$ at long distances. While these models have taught us
much about confinement, they have several features that we do not expect of
the dynamics of _non-Abelian_ confinement. One particularly salient feature
common to all of these models is the complete Higgsing of the $\mathrm{S}_{N}$
subgroup of the $\mathrm{SU}(N)$ gauge group. This Higgsing of
$\mathrm{S}_{N}$ pervades the physics of these theories: it always gives rise
to multiple masses for the dual photons and generically to multiple
fundamental string tensions Douglas:1995nw ; Argyres:1994xh ; Klemm:1994qj
(see Ref. Poppitz:2017ivi  for a case in which fundamental string tensions
remain equal).

The characterization of string tensions is an especially important point of
difference between Abelianizing and non-Abelianizing confining gauge theories.
Indeed, it is well-known that at asymptotically large distances, the string
tensions of confining gauge theories that do not undergo Abelianization should
be solely characterized by center symmetry. This is not the case in the
Abelianizing theories mentioned above, at least within the low-energy
effective field theory, where string tensions are dictated by charges under
$\mathrm{U}(1)^{N-1}$ rather than $N$-ality Douglas:1995nw ; Argyres:1994xh ;
Klemm:1994qj .

Rather curiously, however, it has been observed in numerical experiments that
the dynamics of non-Abelian confinement admit an intermediate distance scale
where the string tensions are not solely characterized by center symmetry
either Ambjorn:1984dp ; Poulis:1995nn ; Bali:2000un ; Philipsen:1999wf ;
Stephenson:1999kh ; deForcrand:1999kr ; Greensite:2003bk ;
Wellegehausen:2010ai : they carry more detailed information on the
representations of the gauge group. This naturally suggests that we should try
to construct an intermediate theory between Abelian and non-Abelian worlds to
leverage what we know about the former to learn more about the latter. As
mentioned above, an understanding of unbroken $\mathrm{S}_{N}$ should be an
important clue in this direction.

Thus, the purpose of this work is to construct a 3-dimensional lattice model
in which these considerations can be addressed quite explicitly. We call it
the _semi-Abelian gauge theory_. To define it, we begin with a pure Abelian
lattice model (henceforth to be referred to as the ‘Abelian model’) with gauge
group $\mathrm{U}(1)^{N-1}$ such that the permutation group $\mathrm{S}_{N}$
is present as a global symmetry. The semi-Abelian theory is then obtained by
gauging this $\mathrm{S}_{N}$, and its gauge group is given by

$G_{\mathrm{gauge}}=\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}.$ (1)

As pure gauge theories are _not_ usually equipped with non-Abelian global
symmetries, the global or local $\mathrm{S}_{N}$ symmetry of these models has
some rather interesting consequences.

We first show that in both models, the mass gap is generated via Polyakov’s
mechanism whereby the proliferation of lattice monopole-instantons results in
Debye screening. Crucially, the unbroken (or un-Higgsed) $\mathrm{S}_{N}$
symmetry implies that the effective potential receives equal contributions
from the monopoles associated with the entire $\mathrm{SU}(N)$ root system,
which in turn leads to exact degeneracy for the $N-1$ dual photon masses. (As
the gauge groups of both theories can be realized as subgroups of
$\mathrm{SU}(N)$, we find it useful to use the language of $\mathrm{SU}(N)$
representations.) This feature sharply contrasts with the mass generation in
the Polyakov model on $\mathbb{R}^{3}$ or in deformed Yang–Mills on
$\mathbb{R}^{3}\times S^{1}$, where the effective potential is sourced at
leading order only by the monopoles associated with the (affine) simple roots
and the $N-1$ dual photon masses are not degenerate.

After studying these properties of local operators, we move on to study
properties of test electric particles, which can be described by the behavior
of Wilson loops. In 3 spacetime dimensions, the Coulomb potential is already
log-confining, but due to the mass gap generated by the monopole-instantons,
the interparticle potential becomes linear-confining. Using the dual
formulation of the Wilson loop, we give a semi-classical formula for the
string tensions within a reasonable ansatz. We then find that there are
infinitely many string tensions. In particular, the semi-Abelian theory
furnishes a unique fundamental string tension.

This, however, raises a puzzle about the string tensions. In order to study
the spectral properties of the confining forces for test quarks, we would like
to have some symmetry that acts nontrivially on the Wilson loops. One is the
well-known center symmetry, which has been recently axiomatized in the
framework of higher-form symmetry Gaiotto:2014kfa ; Sharpe:2015mja . Before
gauging $\mathrm{S}_{N}$, our model has a $\mathrm{U}(1)^{N-1}$ $1$-form
symmetry, which provides sufficiently strong selection rules to support
infinitely many string tensions. But after gauging $\mathrm{S}_{N}$, the
center of the gauge group becomes tiny,

$Z(G_{\mathrm{gauge}})=\mathbb{Z}_{N},$ (2)

so the $1$-form symmetry group becomes $\mathbb{Z}_{N}$, as it is in
$\mathrm{SU}(N)$ Yang–Mills. And as we know from $\mathrm{SU}(N)$ Yang–Mills,
the $\mathbb{Z}_{N}$ $1$-form symmetry can only explain the $N$-ality behavior
of the string tensions at asymptotically large distances. However, the list of
string tensions for the semi-Abelian gauge theory turns out to be unchanged by
the gauging of $\mathrm{S}_{N}$. Thus, as in the case of $\mathrm{SU}(N)$
Yang–Mills at intermediate distances, the string tensions of the semi-Abelian
theory cannot be dictated by $N$-ality alone.

We find a resolution to this puzzle in a not-so-obvious but important symmetry
of the semi-Abelian theory, a non-invertible symmetry. Indeed, after the
generalization of symmetry to higher-form symmetry, it has been recognized
that the most essential feature of a conservation law is the existence of
topological defects, at least in the context of relativistic quantum field
theories (QFTs). In other words, as long as one keeps intact the existence of
topological defect operators, one may make up a new kind of symmetry by
replacing or weakening other features of the generalized global symmetry of
Ref. Gaiotto:2014kfa  (e.g. higher-group symmetry Kapustin:2013uxa ;
Sharpe:2015mja ; Cordova:2018cvg ; Benini:2018reh ; Misumi:2019dwq ;
Tanizaki:2019rbk ; Hidaka:2020iaz ; Hidaka:2020izy ). Non-invertible
symmetries are also generated by topological defects, but their fusion rules
do not conform to the usual group multiplication; as the name suggests, a non-
invertible symmetry transformation need not have an inverse (which of course
never occurs if the transformations form a group). The notion of non-
invertible symmetry is still in its infancy, and it seems that its
mathematical formulation has been so far established only in $2$-dimensional
spacetimes. Nevertheless, the utility of such topological operators in probing
quantum systems has been elucidated in several recent studies, as the notion
of symmetry itself tends to be broadened Bhardwaj:2017xup ; Buican:2017rxc ;
Freed:2018cec ; Chang:2018iay ; Thorngren:2019iar ; Ji:2019jhk ;
Rudelius:2020orz ; Komargodski:2020mxz ; Aasen:2020jwb . In that context, the
new symmetry goes by various names, such as non-invertible symmetry,
categorical symmetry, etc. Here, we would like to emphasize that the non-
invertible symmetry clarifies an important feature of our $3$-dimensional
semi-Abelian gauge theory. Thanks to the simplicity of the model, the symmetry
considerations we propose can be checked against concrete calculation.

We construct a generator of a continuous non-invertible symmetry, and compute
its action on several Wilson loops. By looking at its eigenvalues, we show
that we can distinguish different string tensions even if they correspond to
representations of the same $N$-ality. We also discuss conditions where such
extra selection rules by noninvertible symmetry are lost by the addition of
dynamical electric particles, and we compare them with the standard string-
breaking arguments to check that they are consistent. Finally, as an
application, we discuss an example where the non-invertible symmetry is
explicitly broken to a discrete sub-symmetry, so that even though the number
of string tensions becomes finite, there still remain some string tensions
beyond $N$-ality.

##  2 3d $\mathrm{U}(1)^{N-1}$ lattice gauge theory with $\mathrm{S}_{N}$
global symmetry

There are two basic models that we study in this paper:

  * •

$\text{U}(1)^{N-1}$ Abelian gauge theory with discrete non-Abelian global
symmetry ${\rm S}_{N}$

  * •

$\text{U}(1)^{N-1}\rtimes\mathrm{S}_{N}$ semi-Abelian gauge theory

The second one can be obtained by gauging the $\mathrm{S}_{N}$ 0-form symmetry
of the first one, and the main purpose of this paper is to understand its
properties. To this end, we must first understand the properties of the first
theory, and this is the goal of this section.

###  2.1 Description of the $\mathrm{U}(1)^{N-1}$ lattice gauge theory

The $\text{U}(1)^{N-1}$ lattice gauge model with the $\mathrm{S}_{N}$ global
symmetry can be realized either by a standard Wilson-type formulation
Wilson:1974sk  or by a Villain-type formulation Villain:1974ir . Since these
provide somewhat complementary perspectives, we end up working with both.

####  2.1.1 Villain formulation

To give the Villain formulation, we consider a link field $\bm{A}_{\ell}$
valued in $\mathbb{R}^{N-1}$ and a plaquette field $\bm{n}_{p}$ valued in the
root lattice ${\Gamma}_{\rm r}\subset\mathbb{R}^{N-1}$ of SU($N$). We take for
the action

$S=\frac{1}{4\pi e^{2}}\sum_{p}(\bm{F}_{p}+2\pi\bm{n}_{p})^{2}$ (3)

where $\bm{F}_{p}=(\text{d}\bm{A})_{p}$ is the field-strength and $e$ is the
gauge coupling. The partition function is given by

$Z=\sum_{\\{\bm{n}_{p}\in{\Gamma}_{\rm
r}\\}}\int_{\mathbb{R}^{N-1}}[\text{d}\bm{A}_{\ell}]\,\text{e}^{-S}\,.$ (4)

This theory is invariant under the $0$-form gauge symmetry

$\bm{A}_{\ell}\rightarrow\bm{A}_{\ell}+(\text{d}\bm{\lambda})_{\ell}\,,\qquad\bm{\lambda}_{s}\in\mathbb{R}^{N-1},$
(5)

and the $1$-form gauge symmetry

$\bm{A}_{\ell}\rightarrow\bm{A}_{\ell}+2\pi\bm{\beta}_{\ell}\,,\qquad\bm{n}_{p}\rightarrow\bm{n}_{p}-(\text{d}\bm{\beta})_{p}\,,\qquad\bm{\beta}_{\ell}\in{\Gamma}_{\rm
r}\,.$ (6)

In view of the fact that $\mathbb{R}^{N-1}/(2\pi{\Gamma}_{\rm r})\simeq{\rm
U}(1)^{N-1}$, we see that this indeed defines a ${\rm U}(1)^{N-1}$ gauge
model.

One way to understand this Villain-type formulation Polyakov:1975rs ;
Sulejmanpasic:2019ytl  is to imagine that we had begun with pure
$\mathbb{R}^{N-1}$ gauge theory,

$Z=\int_{\mathbb{R}^{N-1}}[\text{d}\bm{A}_{\ell}]\exp\left(-\frac{1}{4\pi
e^{2}}\sum_{p}\bm{F}_{p}^{2}\right),$ (7)

and then considered gauging the discrete subgroup $2\pi{\Gamma}_{\rm r}$ of
the $\mathbb{R}^{N-1}$ 1-form center symmetry group, which acts according to

$\bm{A}_{\ell}\mapsto\bm{A}_{\ell}+\bm{\theta}_{\ell}\,,\qquad\bm{\theta}_{\ell}\in\mathbb{R}^{N-1}\,,\qquad(\text{d}\bm{\theta})_{p}=0\,.$
(8)

The simplest way to do that is to introduce the discrete ${\Gamma}_{\rm
r}$-valued plaquette field $\bm{n}_{p}$, and then demand that the local
transformations (6) be gauge redundancies. Minimal coupling to the field
$\bm{n}_{p}$ would then produce the action (3) and the partition function (4).

##### Global symmetries:

Let us now discuss the global symmetries of this model. First, as already
noted above, there is a ${\rm U}(1)^{N-1}$ 1-form center symmetry (8), where
the group is ${\rm U}(1)^{N-1}$ rather than $\mathbb{R}^{N-1}$ thanks to the
1-form gauge structure (6).

Importantly, the theory has a discrete non-Abelian 0-form global symmetry,

$\bm{A}_{\ell}\mapsto\Pi\bm{A}_{\ell}\,,\qquad\bm{n}_{p}\mapsto\Pi\bm{n}_{p},$
(9)

under O($N-1$) transformations $\Pi$ that preserve the root lattice
${\Gamma}_{\rm r}$. Such transformations constitute the automorphism group of
the SU($N$) root system, and therefore the symmetry group here is

$G^{[0]}_{\mathrm{global}}=\left\\{\begin{matrix}\mathrm{S}_{N}\rtimes\mathbb{Z}_{2}&\quad(N>2),\\\
\mathrm{S}_{2}\simeq\mathbb{Z}_{2}&\quad(N=2).\end{matrix}\right.$ (10)

The $\mathrm{S}_{N}$ corresponds to the Weyl group of SU($N$), which is
generated by the reflections in the hyperplanes orthogonal to the roots

$\bm{A}_{\ell}\mapsto\bm{A}_{\ell}-\bm{\alpha}(\bm{\alpha}\cdot\bm{A}_{\ell})\,,\qquad\bm{n}_{p}\mapsto\bm{n}_{p}-\bm{\alpha}(\bm{\alpha}\cdot\bm{n}_{p})\,,\qquad\bm{\alpha}\in\Phi,$
(11)

where $\Phi$ is the set of roots for SU($N$). The pair
$(\bm{A}_{\ell},\bm{n}_{p})$ thus transforms in the standard representation
$D_{\rm std}$ of $\mathrm{S}_{N}$, which is the $(N-1)$-dimensional
irreducible representation. Meanwhile, the $\mathbb{Z}_{2}$ is simply
generated by the reflection

$\bm{A}_{\ell}\mapsto-\bm{A}_{\ell}\,,\qquad\bm{n}_{p}\mapsto-\bm{n}_{p},$
(12)

which we may think of as charge conjugation. We note that, for $N=2$, these
two operations are identical.

Note that the existence of the non-Abelian global symmetry (10) is somewhat
unusual for a pure gauge theory. In general, pure gauge theories without
matter fields, either Abelian or non-Abelian, do not possess non-Abelian
global symmetries. In the ${\rm U}(1)^{N-1}$ gauge theory we are considering,
this symmetry is present. The gauging of the permutation group
$\mathrm{S}_{N}$ will generate a genuinely non-Abelian gauge theory, which we
shall investigate.

The basic observables we are concerned with are the Wilson loops, which are
here given by

$W_{\bm{w}}(C)=\exp\left(\text{i}\int_{C}\bm{w}\cdot\bm{A}\right),$ (13)

with $\bm{w}$ in the weight lattice ${\Gamma}_{\rm w}$ of ${\rm{SU}}(N)$. Note
that it is invariance under the 1-form gauge transformations (6) that requires
the electric charge to be a weight. The Wilson lines transform under the
0-form discrete symmetry (9) as

$W_{\bm{w}}(C)\mapsto W_{\Pi^{-1}\bm{w}}(C),$ (14)

and under the 1-form center symmetry (8) as

$W_{\bm{w}}(C)\mapsto
W_{\bm{w}}(C)\exp\left(\text{i}\int_{C}\bm{w}\cdot\bm{\theta}\right).$ (15)

####  2.1.2 Wilson formulation

To construct the $\text{U}(1)^{N-1}$ lattice gauge theory in the Wilson
formulation, we consider $N$ gauge fields $a^{1}_{\ell},\ldots,a^{N}_{\ell}$
and a Lagrange multiplier $v_{\ell}$ which is an integer-valued link-field.
The dynamics is determined by the action

$S_{\rm W}={\beta}\sum_{p}\sum_{i=1}^{N}(1-\cos
f^{i}_{p})-\text{i}\sum_{\ell}\sum_{i=1}^{N}v_{\ell}a^{i}_{\ell},$ (16)

where the $f^{i}_{p}=(\text{d}a^{i})_{p}$ are the field-strengths. In the
partition function, we integrate over $a^{i}_{\ell}\in[0,2\pi]$ and sum over
$v_{\ell}\in\mathbb{Z}$:

$Z=\sum_{\\{v_{\ell}\in\mathbb{Z}\\}}\int_{0}^{2\pi}[\text{d}a^{i}_{\ell}]\text{e}^{-S_{\mathrm{W}}}.$
(17)

In particular, summation over $v_{\ell}$ in the partition function produces
the constraint

$\sum_{i=1}^{N}a^{i}_{\ell}=0\mod 2\pi,$ (18)

so that only $N-1$ of the photons are physical.111 We could integrate out
$v_{\ell}$ and any one of the photon fields. Then after some simple field
redefinitions, we would obtain the action
$S_{\mathrm{W}}=\beta\sum_{p}\sum_{i=1}^{N}(1-\cos({\bm{{\nu}}}_{i}\cdot{\bm{f}}_{p})),$
(19) where the ${\bm{{\nu}}}_{i}$ are the weights of the defining
representation of SU($N$) and ${\bm{f}}_{p}$ is the field-strength of an
($N-1$)-component Abelian gauge field ${\bm{a}}_{\ell}$.

One nice thing about this formulation is that the $\mathrm{S}_{N}$ symmetry is
manifest; it acts simply by permuting the $N$ photons:

$(a^{1}_{\ell},\ldots,a^{N}_{\ell})\mapsto(a^{P(1)}_{\ell},\ldots,a^{P(N)}_{\ell})\,,\qquad
P\in\mathrm{S}_{N}.$ (20)

For now, we shall prefer to work with the Villain form over the Wilson one,
because the former enjoys exact dualities which allow us to analyze the
dynamics most simply. Nevertheless, the two formulations are equivalent at
weak coupling, as we demonstrate in Appendix B. Later on, in Section 3 where
we gauge the $\mathrm{S}_{N}$ global symmetry, we will find the Wilson form
more convenient.

###  2.2 Mass gap and spectrum

In this subsection, we discuss the mass gap of the lattice Abelian gauge
theory with $\mathrm{S}_{N}$ global symmetry.

First, as we shall review in Section 2.2.1, we note that the Villain form is
exactly dual to a multi-component Coulomb gas; that is, the partition function
can rewritten in the form

$Z=\sum_{\\{\bm{q}(\tilde{x})\in{\Gamma}_{\text{r}}\\}}\exp\left(-\frac{\pi}{e^{2}}\sum_{\tilde{x},\tilde{x}^{\prime}}v(\tilde{x}-\tilde{x}^{\prime})\bm{q}(\tilde{x})\cdot\bm{q}(\tilde{x}^{\prime})\right),$
(21)

where $v(\tilde{x})$ is the lattice Coulomb potential, and $\bm{q}(\tilde{x})$
is a ${\Gamma}_{\rm r}$-valued scalar field on the dual lattice. Here, one can
interpret $\\{\bm{q}({\tilde{x}})\\}$ as a configuration of magnetic
monopoles; $\bm{q}(\tilde{x})$ is the magnetic charge of the magnetic monopole
at $\tilde{x}$. As is familiar, the proliferation of monopoles in the
Euclidean description of the vacuum results in Debye screening, and hence, the
correlation length remains finite for any nonzero value of the coupling
Polyakov:1975rs . While this is more or less self-evident, we can go further
and obtain the long-distance effective field theory:

$Z=\int{\mathcal{D}}\bm{\sigma}\exp\left(-\frac{e^{2}}{2\pi}\int\text{d}^{3}x\,\left\\{\frac{1}{2}|\text{d}\bm{\sigma}|^{2}+M^{2}\sum_{\bm{\alpha}\in\Phi^{+}}\bigl{(}1-\cos(\bm{\alpha}\cdot\bm{\sigma})\bigr{)}\right\\}\right),$
(22)

where the dual photon field $\bm{{\sigma}}$ is a
$2\pi{\Gamma}_{\mathrm{w}}$-periodic scalar, $\Phi^{+}$ is a set of positive
roots for SU($N$), and $M^{2}\propto\text{e}^{-\text{const.}/e^{2}}/e^{2}$.
This effective description shows very clearly the presence of a nonzero mass
gap. It will be derived in Section 2.2.2.

We can immediately observe that the $N-1$ dual photons must have exactly the
same mass. The degeneracy is a consequence of the $\mathrm{S}_{N}$ global
symmetry inherited from the microscopic theory. To see this, note that the
dual photons $\bm{\sigma}$ transform in the standard representation $D_{\rm
std}$ of $\mathrm{S}_{N}$:

$\bm{{\sigma}}\mapsto D_{\rm std}(P)\bm{{\sigma}}\,,\qquad
P\in\mathrm{S}_{N}.$ (23)

The mass matrix for the dual photons,

$(M^{2}_{\bm{\sigma}})^{ij}=M^{2}\sum_{\bm{{\alpha}}\in\Phi^{+}}{\alpha}^{i}{\alpha}^{j},$
(24)

is also invariant under the $\mathrm{S}_{N}$ transformation,

$D_{\rm std}(P)M^{2}_{\bm{\sigma}}D_{\rm
std}^{-1}(P)=M^{2}_{\bm{\sigma}},\qquad P\in\mathrm{S}_{N}.$ (25)

Since $D_{\rm std}$ is irreducible, it follows from Schur’s lemma that
$M^{2}_{\bm{\sigma}}$ must be proportional to the identity matrix. The mass
gap is thus the $(N-1)$-fold degenerate eigenvalue of $M_{\bm{{\sigma}}}^{2}$.
By taking the trace of $M_{\bm{{\sigma}}}^{2}$ and using
$\bm{{\alpha}}^{2}=2$, one easily finds the mass gap to be

$M_{\mathrm{gap}}=\sqrt{N}M\propto\frac{\sqrt{N}}{e}\text{e}^{-\text{const.}/e^{2}}.$
(26)

####  2.2.1 Multi-component Coulomb gas representation of the Villain form

Here we show that the Villain form (4) of our theory is exactly dual to multi-
component Coulomb gas (21), using standard techniques in Abelian lattice gauge
theory Polyakov:1987ez ; Banks:1977cc ; Savit:1977fw ; Gopfert:1981er . We
derive the equivalence very briefly here, but the detailed derivation for the
single-component $\rm{U}(1)$ gauge theory is reviewed in Appendix A.2.

We first note that the Poisson summation formula can be generalized on the
weight and root lattices to give

$\sum_{\bm{n}_{p}\in\Gamma_{\mathrm{r}}}\exp\left(-\frac{1}{4\pi
e^{2}}(\bm{F}_{p}+2\pi\bm{n}_{p})^{2}\right)=\sum_{\bm{k}_{p}\in\Gamma_{\mathrm{w}}}\exp\left(-\pi
e^{2}\bm{k}_{p}^{2}+\mathrm{i}\bm{k}_{p}\cdot\bm{F}_{p}\right)$ (27)

up to an overall coefficient. By performing the ${\bm{A}}_{\ell}$ integration
exactly, we obtain the constraint $(\mathrm{d}^{\dagger}\bm{k})_{\ell}=0,$
which can be easily solved by setting

$*\bm{k}=\mathrm{d}\bm{m},$ (28)

where $\bm{m}(\tilde{x})$ is a $\Gamma_{\mathrm{w}}$-valued scalar field on
the dual lattice.222 For clarity, we ignore the effect of nontrivial spacetime
topology. After this replacement, the partition function becomes

$Z=\sum_{\\{\bm{m}(\tilde{x})\in\Gamma_{\mathrm{w}}\\}}\exp\left(-\pi
e^{2}\sum_{\tilde{\ell}}(\mathrm{d}\bm{m})_{\tilde{\ell}}^{2}\right).$ (29)

We now wish to replace $\bm{m}(\tilde{x})$ by a continuous field; it can be
done with the help of the Poisson summation formula again, this time in the
form

$\sum_{\bm{m}(\tilde{x})\in\Gamma_{\mathrm{w}}}\delta(\bm{\sigma}(\tilde{x})-2\pi\bm{m}(\tilde{x}))=\sum_{\bm{q}(\tilde{x})\in\Gamma_{\mathrm{r}}}\exp(\mathrm{i}\,\bm{q}(\tilde{x})\cdot\bm{\sigma}(\tilde{x})),$
(30)

which introduces the dual photon field $\bm{{\sigma}}(\tilde{x})$. The result
is

$Z=\int[\text{d}\bm{\sigma}(\tilde{x})]\sum_{\\{\bm{q}(\tilde{x})\in\Gamma_{\mathrm{r}}\\}}\exp\left(-\frac{e^{2}}{4\pi}\sum_{\tilde{x}}(\partial_{\mu}^{-}\bm{\sigma}(\tilde{x}))^{2}+\mathrm{i}\sum_{\tilde{x}}\bm{q}(\tilde{x})\cdot\bm{\sigma}(\tilde{x})\right).$
(31)

After performing the Gaussian integration over $\bm{\sigma}$, we arrive at the
multi-component Coulomb gas representation (21).

####  2.2.2 Long-distance effective theory

We now wish to pass to the long-distance effective description (22)
Polyakov:1987ez ; Banks:1977cc ; Savit:1977fw ; Gopfert:1981er . To this end,
we first split the Green function ${\Delta}^{-1}$ in (21) into two parts by
adding and subtracting $(\Delta+M_{\mathrm{PV}}^{2})^{-1}$:

${\Delta}^{-1}={\Delta}^{-1}(1+{\Delta}/M_{\mathrm{PV}}^{2})^{-1}+({\Delta}+M_{\mathrm{PV}}^{2})^{-1}=u_{M_{\rm
PV}}+w_{M_{\rm PV}}.$ (32)

Here, $u_{M_{\mathrm{PV}}}(\tilde{x})$ is the Green function of the
Pauli–Villars regulated Laplacian
$\Delta_{M_{\mathrm{PV}}}\equiv{\Delta}(1+{\Delta}/M_{\mathrm{PV}}^{2})$, and
$w_{M_{\mathrm{PV}}}(\tilde{x})$ is the Yukawa Green function. Since
$w_{M_{\mathrm{PV}}}(\tilde{x})$ decays exponentially fast, we can take
$w_{M_{\mathrm{PV}}}(\tilde{x})=w_{M_{\mathrm{PV}}}(0)\delta_{\tilde{x},0}$.
Furthermore, it is straightforward to show that
$w_{M_{\mathrm{PV}}}(0)=v(0)-{\cal O}(1/M_{\mathrm{PV}})\approx 0.253-{\cal
O}(1/M_{\mathrm{PV}})$ Ukawa:1979yv .

With the above decomposition, we rewrite the Coulomb gas partition function as

$Z=\sum_{\\{\bm{q}(\tilde{x})\in{\Gamma}_{\text{r}}\\}}\exp\left(-\frac{\pi}{e^{2}}\sum_{\tilde{x},\tilde{x}^{\prime}}u_{M_{\mathrm{PV}}}(\tilde{x}-\tilde{x}^{\prime})\bm{q}(\tilde{x})\cdot\bm{q}(\tilde{x}^{\prime})-\frac{1}{2}I\sum_{\tilde{x}}\bm{q}(\tilde{x})^{2}\right),$
(33)

where $I\equiv 2\pi v(0)/e^{2}$, and then reintroduce the dual photon field
$\bm{{\sigma}}$ to get

$Z=\int[\text{d}\bm{{\sigma}}(\tilde{x})]\text{e}^{-\frac{e^{2}}{4\pi}\sum_{\tilde{x}}\bm{{\sigma}}(\tilde{x})\Delta_{M_{\mathrm{PV}}}\bm{{\sigma}}(\tilde{x})}\sum_{\\{\bm{q}(\tilde{x})\in{\Gamma}_{\text{r}}\\}}\text{e}^{\text{i}\sum_{\tilde{x}}\bm{q}(\tilde{x})\bm{{\sigma}}(\tilde{x})}\text{e}^{-\frac{1}{2}I\sum_{\tilde{x}}\bm{q}(\tilde{x})^{2}}.$
(34)

At this point, we want to perform a cluster expansion of the partition
function. For weak coupling, $I$ is large, and so $\text{e}^{-I}$ is
exponentially small. Thus, at leading order in semi-classics, we can restrict
the summation over $\bm{q}(\tilde{x})$ to $\\{0\\}\cup\Phi$. Indeed,
${\bm{{\alpha}}}^{2}=2$ for each $\bm{{\alpha}}\in\Phi$, so all the monopoles
whose charges are roots have the same minimal action $I$, and there are
$N(N-1)$ degenerate saddles at leading order in semi-classics. Performing the
summation over ${\bm{q}}(\tilde{x})$ with this restriction then yields

$\sum_{\bm{q}(\tilde{x})\in\\{0\\}\cup\Phi}\text{e}^{\text{i}\bm{q}(\tilde{x})\bm{{\sigma}}(\tilde{x})}\text{e}^{-\frac{1}{2}I\bm{q}(\tilde{x})^{2}}\approx\exp\left(2\text{e}^{-I}\sum_{\bm{{\alpha}}\in\Phi^{+}}\cos(\bm{{\alpha}}\cdot\bm{{\sigma}}(\tilde{x}))\right).$
(35)

Finally, inserting this into (34), we get

$Z=\int[\text{d}\bm{{\sigma}}(\tilde{x})]\exp\left(-\frac{e^{2}}{4\pi}\sum_{\tilde{x}}\bm{{\sigma}}(\tilde{x})\Delta_{M_{\mathrm{PV}}}\bm{{\sigma}}(\tilde{x})+2\text{e}^{-I}\sum_{\tilde{x}}\sum_{\bm{{\alpha}}\in\Phi^{+}}\cos(\bm{{\alpha}}\cdot\bm{{\sigma}}(\tilde{x}))\right),$
(36)

which, upon taking the continuum limit, coincides with (22).

We note that, in this derivation, we have neglected the effect of the
spacetime topology, and thus the periodicity of the dual photon field
$\bm{\sigma}$ is undetermined. Had we taken it into account, we would have
identified $\bm{\sigma}$ as a $2\pi\Gamma_{\mathrm{w}}$-periodic scalar. (We
elaborate on this subtlety in Appendix A.2.)

##### Remarks:

The fact that the sum over monopoles goes over all roots
${\bm{{\alpha}}}\in\Phi$ and that all monopoles associated with these roots
have the same action distinguishes our ${\rm U}(1)^{N-1}$ lattice gauge theory
with $\mathrm{S}_{N}$ symmetry from Yang–Mills adjoint Higgs systems which
exhibit dynamical Abelianization $\mathrm{SU}(N)\rightarrow{\rm
U}(1)^{N-1}$.444 It is also worth nothing that in $\mathcal{N}=4$
$\mathrm{SU}(N)$ super Yang–Mills theory softly broken down to
$\mathcal{N}=1^{*}$ on $\mathbb{R}^{3}\times S^{1}$ as well, it is necessary
to sum over monopoles associated with non-simple roots in order to capture the
ground state properties correctly Dorey:1999sj . This data is encoded in an
elliptic superpotential, but the $\mathrm{S}_{N}$ symmetry is still Higgsed in
generic vacua. In the latter, if the adjoint Higgs are algebra-valued, as in
the Polyakov model Polyakov:1975rs , the sum over monopoles at leading order
in semi-classics is restricted to the $N-1$ simple roots
${\bm{{\alpha}}}\in\Delta$, while if the adjoint Higgs are group-valued, as in
deformed Yang–Mills Unsal:2008ch , the sum over monopoles is restricted to the
$N$ affine simple roots. There are monopoles associated with non-simple roots
as well, but these are higher action and do not contribute at leading order;
in general, the monopoles split into $\mathbb{Z}_{N}$-orbits with hierarchical
fugacities
${\text{e}}^{-S_{0}}\gg{\text{e}}^{-2S_{0}}\gg\cdots\gg{\text{e}}^{-(N-1)S_{0}}$.
In our present construction, $\mathrm{S}_{N}$ permutation symmetry guarantees
that all $N(N-1)$ monopoles associated with the roots have the same action. In
theories like the Polyakov model and Seiberg–Witten theory, $\mathrm{S}_{N}$
is part of the gauge structure of the microscopic theory, but it is
spontaneously broken by the vacuum expectation value of the Higgs field which
imposes an ordering on the eigenvalues of the adjoint Higgs. These models
therefore exhibit $\mathcal{O}(N)$ different types of fundamental string
tensions. We will see how the string tensions behave in our
$\mathrm{U}(1)^{N-1}$ Abelian model in the following subsection.

###  2.3 Wilson loops and string tensions

In this subsection, we show that the Abelian gauge model confines and we
approximately determine the string tensions.

We begin by showing how a Wilson loop
$W_{\bm{w}}(C)=\exp\left(\mathrm{i}\int_{C}\bm{w}\cdot\bm{A}\right)$ with
electric charge ${\bm{w}}\in{{\Gamma}}_{\rm w}$ is computed in the long-
distance effective theory Polyakov:1987ez . For our purposes, it will suffice
to take $C$ to be a contractible loop, so that it is the boundary of a
2-dimensional surface $D$. We can then write

$W_{\bm{w}}(C)=\exp\left(\mathrm{i}\int_{D}\bm{w}\cdot\bm{F}\right)=\exp\left(\mathrm{i}\sum_{p}[D]_{*p}(\bm{w}\cdot\bm{F}_{p})\right).$
(37)

Here we have introduced the Poincaré dual $[D]$ of $D$; it is a bump $1$-form
on the dual lattice (see Figure 1) given by

$[D]_{*p}=\begin{cases}1,\qquad\text{if}\,\,\,p\subset D,\\\
0,\qquad\text{otherwise}.\end{cases}$ (38)

Figure 1: Left: The dual of the plaquette $p$ is the link $*p$ on the dual
lattice intersecting $p$ as shown. Right: $D$ is the shaded region bounded by
the curve $C$. The Poincare dual $[D]$ is a bump 1-form on the dual lattice
that is 1 on each of the red links shown, and 0 everywhere else.

Now let us repeat the derivation of the dual theory, this time with the
insertion of the Wilson loop. Using the Poisson resummation (27), the path-
integral weight becomes

$\exp\left(-\pi
e^{2}\bm{k}_{p}^{2}+\mathrm{i}\bm{k}_{p}\cdot\bm{F}_{p}+\mathrm{i}[D]_{*p}(\bm{w}\cdot\bm{F}_{p})\right),$
(39)

where the last term comes from the Wilson loop. The integration over $\bm{A}$
produces the constraint, $\mathrm{d}^{\dagger}(\bm{k}+*(\bm{w}[D]))=0$, which
can be solved by

$*\bm{k}=-\bm{w}[D]+\mathrm{d}\bm{m},$ (40)

instead of (28). At this point, the rest of the derivation proceeds exactly as
before, and we obtain

$\langle
W_{\bm{w}}(C)\rangle=\int{\mathcal{D}}\bm{\sigma}\exp\left(-\frac{e^{2}}{2\pi}\int\text{d}^{3}x\,\left\\{\frac{1}{2}\Bigl{|}\mathrm{d}\bm{\sigma}-2\pi\bm{w}[D]\Bigr{|}^{2}+M^{2}\sum_{\bm{\alpha}\in\Phi^{+}}\bigl{(}1-\cos(\bm{\alpha}\cdot\bm{\sigma})\bigr{)}\right\\}\right).$
(41)

This expression shows that the Wilson loop is realized as a defect operator in
the dual formulation. That is to say, it is evaluated by removing the loop $C$
from the spacetime and restricting the path integral to configurations
satisfying $\oint_{S^{1}}\mathrm{d}\bm{\sigma}=2\pi\bm{w}$ for small loops
$S^{1}$ that link with the loop $C$. Taking the ratio with the unconstrained
path integral, we obtain the expectation value of the Wilson loop.

We are now in a position to approximately determine the string tensions. It
will suffice to compute the functional integral in (41) in the classical
approximation. For convenience, let us take the loop $C$ as well as $D$ to lie
in the $z\equiv x_{3}=0$ plane. If we take the loop $C$ to be so large that
$D$ essentially fills the $z=0$ plane, then the action density localizes
around $z=0$. As a result, the area law decay for the Wilson loop
$W_{\bm{w}}(C)$ is observed,

$\langle
W_{\bm{w}}(C)\rangle\sim\exp\Bigl{(}-T_{\bm{w}}\,\mathrm{Area}(D)\Bigr{)},$
(42)

and its string tension is given by the minimal action density

$T_{\bm{w}}=\min_{\bm{\sigma}(z)}\,\frac{e^{2}}{2\pi}\int_{-\infty}^{+\infty}\text{d}z\,\left\\{\frac{1}{2}\left(\frac{\text{d}\bm{\sigma}}{\text{d}z}\right)^{2}+M^{2}\sum_{\bm{\alpha}\in\Phi^{+}}\Bigl{(}1-\cos(\bm{\alpha}\cdot\bm{\sigma})\Bigr{)}\right\\},$
(43)

with the boundary condition

$\bm{\sigma}(-\infty)=0,\qquad\bm{\sigma}(+\infty)=2\pi\bm{w}.$ (44)

To go further, let us take as a plausible ansatz

$\bm{\sigma}(x)=\bm{w}\sigma(z),$ (45)

and then proceed to evaluate the string tension analytically within this
ansatz. Substituting (45) into (43), we obtain

$\displaystyle T_{\bm{w}}$ $\displaystyle=$
$\displaystyle\min_{{\sigma}(z)}\,\frac{e^{2}}{2\pi}\int_{-\infty}^{+\infty}\text{d}z\,\left\\{\frac{\bm{w}^{2}}{2}\left(\frac{\text{d}\sigma}{\text{d}z}\right)^{2}+M^{2}\sum_{\bm{\alpha}\in\Phi^{+}}\Bigl{(}1-\cos(\bm{\alpha}\cdot\bm{w}\sigma)\Bigr{)}\right\\}$
(46) $\displaystyle=$
$\displaystyle\frac{e^{2}M}{\pi}\int_{0}^{2\pi}\text{d}\sigma\,\sqrt{\frac{\bm{w}^{2}}{2}\sum_{\bm{\alpha}\in\Phi^{+}}\Bigl{(}1-\cos(\bm{\alpha}\cdot\bm{w}\sigma)\Bigr{)}},$

which is the Bogomol’nyi–Prasad–Sommerfield (BPS) bound Bogomolny:1975de ;
Prasad:1975kr . Although this is just an upper bound for the actual string
tension, we assume that it gives a reasonable estimate.

####  2.3.1 Explicit evaluation of string tensions

Using the formula (46), we shall evaluate the string tensions explicitly for a
few cases. Here, we take $\bm{w}=\bm{\mu}_{1},\,2\bm{\mu}_{1},\,\bm{\mu}_{2}$,
which correspond to the highest weights of the fundamental, symmetric, and
anti-symmetric representations of $\mathrm{SU}(N)$, respectively. We will also
comment on the case $\bm{w}=\bm{\alpha}\in\Gamma_{\mathrm{r}}$, corresponding
to the adjoint representation of $\mathrm{SU}(N)$.

Let us start with $\bm{w}=\bm{\mu_{1}}$, which is the highest weight of the
fundamental representation of $\mathrm{SU}(N)$. We obtain

$\displaystyle T_{\bm{\mu}_{1}}$ $\displaystyle=$
$\displaystyle{e^{2}M\over\pi}{N-1\over\sqrt{2N}}\int_{0}^{2\pi}\mathrm{d}\sigma\sqrt{1-\cos(\sigma)}$
(47) $\displaystyle=$ $\displaystyle{4e^{2}M\over\pi}{N-1\over\sqrt{N}}.$

We will use this quantity as a unit for the other string tensions.

We next consider $\bm{w}=2\bm{\mu}_{1}$, the highest weight of the
$\mathrm{SU}(N)$ two-index symmetric representation. Since $\sigma$ wraps
$S^{1}$ twice, we find that

$T_{2\bm{\mu}_{1}}=2T_{\bm{\mu}_{1}}.$ (48)

Thus, the symmetric string tension is twice the fundamental one, which
suggests that the symmetric string can be interpreted as the sum of two
independent fundamental strings. The multi-string ansatz is also a candidate,
which may give a reasonable approximation of confining strings Anber:2015kea ,
so we will compare it with (45) for other strings, too.

For the two-index anti-symmetric string, $\bm{w}=\bm{\mu}_{2}$, we find

$\displaystyle T_{\bm{\mu}_{2}}$ $\displaystyle=$
$\displaystyle{e^{2}M\over\pi}{\sqrt{2}(N-2)\over\sqrt{N}}\int_{0}^{2\pi}\mathrm{d}\sigma\sqrt{1-\cos(\sigma)}$
(49) $\displaystyle=$ $\displaystyle{8e^{2}M\over\pi}{N-2\over\sqrt{N}}$
$\displaystyle=$ $\displaystyle{2(N-2)\over N-1}T_{\bm{\mu_{1}}}.$

For $N=3$, $\bm{\mu}_{2}$ gives the conjugate representation of
$\bm{\mu}_{1}$, and in this case we indeed find that
$T_{\bm{\mu}_{2}}=T_{\bm{\mu}_{1}}$. For $N>3$, we find
$T_{\bm{\mu}_{1}}<T_{\bm{\mu}_{2}}<T_{2\bm{\mu}_{1}}=2T_{\bm{\mu_{1}}}$, and
$T_{\bm{\mathrm{\mu}_{2}}}\approx 2T_{\bm{\mu}_{1}}$ for $N\gg 1$. Since
$\bm{\mu}_{2}=\bm{\mu}_{1}+(\bm{\mu}_{1}-\bm{\alpha}_{1})$, we can understand
this upper bound $2T_{\bm{\mu}_{1}}$ as a sum of two-independent fundamental
strings $2T_{\bm{\mu}_{1}}=T_{\bm{\mu}_{1}}+T_{\bm{\mu}_{1}-\bm{\alpha}_{1}}$.
Our calculation shows that, for the anti-symmetric string, the ansatz (45)
gives a more severe upper bound for $T_{\bm{\mu}_{2}}$.

Lastly, let us consider the adjoint string $\bm{w}=\bm{\alpha}\in\Phi$.
Applying the formula (46) within the ansatz (45), we obtain it as

$\displaystyle T_{\bm{\alpha}}$ $\displaystyle=$
$\displaystyle{e^{2}M\over\pi}\int_{0}^{2\pi}\mathrm{d}\sigma\sqrt{(1-\cos(2\sigma))+2(N-2)(1-\cos(\sigma))}$
(50) $\displaystyle=$
$\displaystyle{4e^{2}M\over\pi}\left(\sqrt{N}+{(N-2)\over\sqrt{2}}\cosh^{-1}\sqrt{N\over
N-2}\right)$ $\displaystyle=$ $\displaystyle{1\over
N-1}\left(N+{(N-2)\sqrt{N}\over\sqrt{2}}\cosh^{-1}\sqrt{N\over
N-2}\right)T_{\bm{\mu_{1}}}.$

According to this formula, the adjoint string tension satisfies
$T_{\bm{\alpha}}\geq 2T_{\bm{\mu}_{1}}$, and turns out to be only slightly
larger than $2T_{\bm{\mu}_{1}}$. But here it turns out that we can do a little
bit better. Let us explicitly take $\bm{w}=\bm{\alpha}_{1}$, and consider a
double-string ansatz, in which the adjoint string consists of two fundamental
strings, $\bm{\mu}_{1}$ and $\bm{\alpha}_{1}-\bm{\mu}_{1}$. Up to permutation,
$\bm{\mu}_{1}$ and $\bm{\alpha}_{1}-\bm{\mu}_{1}$ are related by complex
conjugation, and we thus find

$T_{\bm{\alpha}}=T_{\bm{\mu}_{1}}+T_{\bm{\alpha}_{1}-{\bm{\mu}_{1}}}=2T_{\bm{\mu}_{1}}.$
(51)

Therefore, for the adjoint string, the two-independent-string ansatz is
slightly better than (45), unlike the case of anti-symmetric string.

##  3 Semi-Abelian theory

In this section, we consider the $\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}$
gauge theory obtained by gauging the $\mathrm{S}_{N}$ global symmetry of the
$\mathrm{U}(1)^{N-1}$ gauge theory considered in Section 2. We call it the
semi-Abelian gauge theory.

###  3.1 Gauging of the $\mathrm{S}_{N}$ global symmetry

In relativistic quantum field theories, global symmetry is generated by a set
of codim-$1$ defects, which are topological and obey the group-multiplication
law Gaiotto:2014kfa . When the global symmetry is discrete, we can gauge it by
summing over all possible networks of such codim-$1$ defects. This procedure
may be obstructed by anomalies, which are characterized by a topological
action in one higher dimension Cho:2014jfa ; Kapustin:2014lwa ;
Kapustin:2014zva . We also note that the gauging procedure admits the freedom
to add a topological phase to each network configuration of the topological
defects as long as it is consistent with locality and unitarity.

In this section, we gauge the $\mathrm{S}_{N}$ symmetry of the
$\mathrm{U}(1)^{N-1}$ lattice gauge theory of Section 2. Assuming that its
low-energy description enjoys emergent Lorentz symmetry due to the cubic
lattice rotational invariance, the gauging procedure of $\mathrm{S}_{N}$
should fit into the above general discussion. Absence of the $\mathrm{S}_{N}$
anomaly is guaranteed by the explicit construction of the lattice gauge
theory. As extra topological terms, there are Dijkgraaf-Witten (DW) terms
Dijkgraaf:1990nc  characterized by $H^{3}(B\mathrm{S}_{N},\mathrm{U}(1))$,
which are nontrivial for all $N\geq 2$.555Although detailed information is not
relevant for us as we neglect the nontrivial DW twist, let us give its full
information for completeness, which may be useful for possible extensions. By
the universal coefficient theorem, we obtain
$H^{d}(B\mathrm{S}_{N},\mathrm{U}(1))\simeq
H^{d+1}(B\mathrm{S}_{N},\mathbb{Z})\simeq H_{d}(B\mathrm{S}_{N},\mathbb{Z})$
because they only have the torsion part. We can find in literatures that the
list of the 3d DW twist is given as
$\begin{array}[]{c|cccccc}N&2&3&4&5&6&\cdots\\\ \hline\cr
H^{3}(B\mathrm{S}_{N},\mathrm{U}(1))&\,\,\mathbb{Z}_{2}&\,\mathbb{Z}_{6}&\mathbb{Z}_{2}\oplus\mathbb{Z}_{12}&\mathbb{Z}_{2}\oplus\mathbb{Z}_{12}&(\mathbb{Z}_{2})^{2}\oplus\mathbb{Z}_{12}&\cdots\end{array}$
(52) For $N\geq 6$, this group cohomology stabilizes and
$H^{3}(B\mathrm{S}_{N},\mathrm{U}(1))\simeq(\mathbb{Z}_{2})^{2}\oplus\mathbb{Z}_{12}$,
i.e. we can add three distinct DW terms, two of which give the $(\pm 1)$
phases and the another one gives the phases $\exp\left({2\pi\mathrm{i}\over
12}n\right)$, in the path integral of $\mathrm{S}_{N}$ gauge fields.  In this
paper, we limit ourselves to the case without the $3$d $\mathrm{S}_{N}$ DW
term.

We shall now construct the semi-Abelian gauge theory on a cubic lattice, and
the simplest way to proceed is to gauge $\mathrm{S}_{N}$ in the Wilson
formulation (16). For concreteness, it is convenient to realize the semi-
Abelian gauge symmetry by $N\times N$ matrices. We can realize an element of
$\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}$ inside $\mathrm{SU}(N)$ as

$P\cdot C\in\mathrm{SU}(N),$ (53)

where
$C=\mathrm{diag}(\mathrm{e}^{\mathrm{i}a_{1}},\cdots,\mathrm{e}^{\mathrm{i}a_{N}})$
with $\det(C)=1$ describes the Cartan components, and $P\in\mathrm{S}_{N}$ is
the $N\times N$ matrix representation of a Weyl reflection, which is realized
as a signed permutation matrix. The group multiplication law is given as

$(P_{1}\cdot C_{1})(P_{2}\cdot
C_{2})=\underbrace{(P_{1}P_{2})}_{\in\mathrm{S}_{N}}\cdot\underbrace{((P_{2}^{-1}C_{1}P_{2})C_{2})}_{\in\mathrm{U}(1)^{N-1}},$
(54)

and this expression elucidates the semi-direct structure in a concrete manner.
Letting $(P_{\ell}\cdot C_{\ell})\in\mathrm{SU}(N)$ denote the link variable,
the gauge-invariant plaquettes for the Lagrangian consist of two terms:

$\mathrm{Re}\left(\beta_{1}{\rm tr}\left[\bm{1}_{N}-\prod_{\ell\subset\partial
p}(P_{\ell}\cdot C_{\ell})\right]+\beta_{2}{\rm
tr}\left[\bm{1}_{N}-\prod_{\ell\subset\partial p}P_{\ell}\right]\right).$ (55)

The second term is the gauge-invariant kinetic term only for $\mathrm{S}_{N}$.
By sending $\beta_{2}\to+\infty$, we can impose the flatness condition on the
$\mathrm{S}_{N}$ gauge field,

$\prod_{\ell\subset\partial p}P_{\ell}=\bm{1}_{N}\in\mathrm{S}_{N}.$ (56)

As it is this limit that fits into the general discussion given above for the
continuum description, we shall work with this flatness condition (56) on the
$\mathrm{S}_{N}$ link variables. We can readily check that the Lagrangian (55)
is equal to the Wilson action (16) when the $\mathrm{S}_{N}$ gauge fields are
trivial, i.e. $P_{\ell}=1$ for all the links $\ell$, by putting
$\beta_{1}={\beta}$. In this manner, we obtain the
$\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}$ gauge theory out of the
$\mathrm{U}(1)^{N-1}$ pure gauge theory.

Because of the flatness condition (56), the local dynamics should not be much
affected by the gauging of $\mathrm{S}_{N}$, and all the interesting things in
the deep infrared have to do with the global aspects of the theory. To be more
precise, let us assume that we prepare a sufficiently large torus $T^{3}$ for
the spacetime and that we are interested in computing correlation functions
inside an open ball $B^{3}\subset T^{3}$, which has a trivial topology. Using
the flatness condition, we may perform a local $\mathrm{S}_{N}$ gauge
transformation so that the $\mathrm{S}_{N}$ gauge fields $P_{\ell}$ are fixed
to equal 1 inside $B^{3}$. Hence, the correlation functions should be
identical with those of the $\mathrm{U}(1)^{N-1}$ theory of Section 2 as it
has a nonzero mass gap, as long as we neglect exponentially small corrections
that vanish in the thermodynamic limit. In this sense, the gauging of
$\mathrm{S}_{N}$ is locally trivial.

More physically, by sending the parameter $\beta_{2}\to\infty$ in the
Lagrangian, we make the magnetic monopoles for the $\mathrm{S}_{N}$ gauge
group extremely heavy. As a result, the $\mathrm{S}_{N}$ gauge fields are
deconfined; i.e. the Wilson loops of $\mathrm{S}_{N}$ gauge fields obey the
perimeter law at any length scale. Now, assume that we wish to compute the
correlation functions of local operators. For the sake of exposition, consider
a two-point function of the $\mathrm{U}(1)^{N-1}$ gauge theory

$\left\langle O_{1}(x_{1})O_{2}(x_{2})\right\rangle_{\mathrm{U}(1)^{N-1}},$
(57)

though it is straightforward to extend the discussion to general $n$-point
functions. As the $\mathrm{S}_{N}$ global symmetry is not spontaneously broken
in the $\mathrm{U}(1)^{N-1}$ theory, this correlation function has a non-zero
expectation value in the thermodynamic limit only if
$O_{1}(x_{1})O_{2}(x_{2})$ contains an $\mathrm{S}_{N}$-singlet component.
Thus, we may assume that $O_{1}(x_{1})O_{2}(x_{2})$ is $\mathrm{S}_{N}$
singlet without loss of generality. Here, we note that the operator $O_{1}$,
$O_{2}$ can be $\mathrm{S}_{N}$ non-singlet, but they have to be mutually
conjugate representations. By introducing an $\mathrm{S}_{N}$ Wilson line
$W^{(\mathrm{S}_{N})}(x_{1},x_{2})$ connecting $x_{1}$ and $x_{2}$, we can
construct the $\mathrm{S}_{N}$ gauge invariant operator
$O_{1}(x_{1})W^{(\mathrm{S}_{N})}(x_{1},x_{2})O_{2}(x_{2})$. Since the
$\mathrm{S}_{N}$ gauge field is deconfined, we find that $\left\langle
O_{1}(x_{1})W^{(\mathrm{S}_{N})}(x_{1},x_{2})O_{2}(x_{2})\right\rangle_{\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}}$
is invariant under any continuous deformation of the path connecting $x_{1}$
and $x_{2}$. In particular, by choosing a trivial path which does not go
around any nontrivial cycle of the spacetime, we obtain

$\left\langle
O_{1}(x_{1})W^{(\mathrm{S}_{N})}(x_{1},x_{2})O_{2}(x_{2})\right\rangle_{\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}}=\left\langle
O_{1}(x_{1})O_{2}(x_{2})\right\rangle_{\mathrm{U}(1)^{N-1}}$ (58)

in the thermodynamic limit. Thus, any local correlation function of the
$\mathrm{U}(1)^{N-1}$ theory can be recovered in the semi-Abelian gauge
theory.

###  3.2 $\mathbb{Z}_{N}$ center symmetry

In this and the following subsections, we discuss properties of the (electric)
Wilson loops in order to identify the string tensions from the viewpoint of
the symmetry. Here, we pay especial attention to the $1$-form symmetry, or
center symmetry, of the $\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}$ gauge
theory.

The $1$-form symmetry is generated by codim-$2$ topological defects, whose
fusion rule obeys group multiplication Gaiotto:2014kfa . In general, when we
consider a pure gauge theory with a gauge group $G_{\mathrm{gauge}}$, the
theory enjoys a $1$-form symmetry with the symmetry group
$Z(G_{\mathrm{gauge}})$, which is the center of $G_{\mathrm{gauge}}$. Since
this acts as $Z(G_{\mathrm{gauge}})$ phase rotations on Wilson loops, this has
been historically called the center symmetry.

In our case, the gauge group is
$G_{\mathrm{gauge}}=\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}$, and thus

$Z(G_{\mathrm{gauge}})=\mathbb{Z}_{N}.$ (59)

To see this, it is convenient to use the embedding of
$\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}\subset\mathrm{SU}(N)$ used above,
and to consider the defining representation of the latter. Using Schur’s
lemma, one sees that the $N\times N$ matrix representation of center elements
must be proportional to the identity matrix. Such matrices are included only
in $\mathrm{U}(1)^{N-1}$, which is the same as the Cartan factor of
$\mathrm{SU}(N)$, and thus the center elements of
$\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}$ are the same as those of
$\mathrm{SU}(N)$. Before gauging $\mathrm{S}_{N}$, the $1$-form symmetry group
is given by $Z(\mathrm{U}(1)^{N-1})=\mathrm{U}(1)^{N-1}$ since the theory is
an Abelian gauge theory without any electric matter fields. Therefore, in view
of the $1$-form symmetry, one might be led to claim that the semi-Abelian
gauge theory should be more similar to $\mathrm{SU}(N)$ gauge theories than
the $\mathrm{U}(1)^{N-1}$ theory from which it came.

This, however, raises the following puzzle about the string tensions. In the
$\mathrm{U}(1)^{N-1}$ theory, there are infinitely many different string
tensions depending on the representations of Wilson loops, which are
characterized by charges of the $\mathrm{U}(1)^{N-1}$ $1$-form symmetry. As we
have seen in the previous subsection, the local dynamics is not affected by
the gauging of $\mathrm{S}_{N}$ because we can locally set the
$\mathrm{S}_{N}$ gauge field to be zero by gauge transformations. As long as
we measure string tensions using large and contractible Wilson loops, the same
discussion from before should apply here, and thus, there have to be
infinitely many different string tensions also for the
$\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}$ gauge theory. But this seems rather
unnatural, because the $\mathbb{Z}_{N}$ $1$-form symmetry is too weak to give
selection rules for these string tensions. To put the question another way:

> _How is the presence of infinitely many different string tensions compatible
> with the finite center symmetry?_

To make things more concrete, let us construct the generator of
$\mathbb{Z}_{N}^{[1]}$ out of $(\mathrm{U}(1)^{[1]})^{N-1}$ generators of the
$\mathrm{U}(1)^{N-1}$ gauge theory. In terms of the dual photon field
$\bm{\sigma}\in\mathbb{R}^{N-1}/2\pi\Gamma_{w}$ from the monopole gas
description in Section 2.2, the generators of $(\mathrm{U}(1)^{[1]})^{N-1}$
are given by

$U_{\theta}^{(k)}(C)=\exp\left(\mathrm{i}{\theta\over
2\pi}\int_{C}\bm{\alpha}_{k}\cdot\mathrm{d}\bm{\sigma}\right),\qquad
k=1,\ldots,N-1,$ (60)

where the transformation parameter $\theta$ is $2\pi$ periodic due to the
$2\pi{\Gamma}_{\mathrm{w}}$-periodicity of $\bm{\sigma}$. After gauging
$\mathrm{S}_{N}$, these operators are no longer gauge invariant because the
dual photon field $\bm{\sigma}$ transforms under the standard representation
of the Weyl permutations $\mathrm{S}_{N}$. Nevertheless, the generators of
$\mathbb{Z}_{N}^{[1]}$ can be constructed as

$U_{n}(C)\equiv\prod_{k=1}^{N-1}U_{{2\pi\over
N}kn}^{(k)}(C)=\exp\left(\mathrm{i}{n\over
N}\int_{C}(\bm{\alpha}_{1}+2\bm{\alpha}_{2}+\cdots+(N-1)\bm{\alpha}_{N-1})\cdot\mathrm{d}\bm{\sigma}\right).$
(61)

Thanks to the periodicity of $\bm{\sigma}$, the $U_{n}(C)$ are invariant under
$\mathrm{S}_{N}$ transformations and thus remain good operators for the semi-
Abelian gauge theory. Moreover, we have the group multiplication law

$U_{n}(C)U_{m}(C)=U_{n+m\;{\rm mod\;}N}(C)$ (62)

so $U(C)\equiv U_{1}(C)$ only generates the $\mathbb{Z}_{N}$ subgroup of
$\mathrm{U}(1)^{N-1}$. Other generic combinations
$\prod_{k}U_{\theta_{k}}^{(k)}$ cannot satisfy the $\mathrm{S}_{N}$
invariance, and they drop out from the possible generators of the $1$-form
symmetry.

###  3.3 String tensions beyond $N$-ality, and noninvertible topological lines

Let us explicitly check whether or not the string tensions of the semi-Abelian
gauge theory obey the standard $N$-ality rule. Using the embedding
$\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}\subset\mathrm{SU}(N)$, we construct
the Wilson loops with the $\mathrm{SU}(N)$ gauge field first, and then we
restrict it to the $\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}$ gauge field. As
we can locally eliminate the $\mathrm{S}_{N}$ gauge field, we may restrict the
$\mathrm{SU}(N)$ gauge field to its diagonal component in a naive way, as long
as the Wilson loop is contractible. The $N$-ality of the obtained Wilson loop
is the same as that of the Wilson loop with $\mathrm{SU}(N)$ gauge fields.

Let us write

$W_{k}(C)=\exp\left(\mathrm{i}\Bigl{(}\bm{\mu}_{1}-\sum_{j=1}^{k-1}\bm{\alpha}_{j}\Bigr{)}\cdot\int_{C}\bm{A}\right),\qquad
k=1,\ldots,N,$ (63)

so that, for example, the fundamental Wilson loop is given by

$W_{\mathrm{fd}}(C)=W_{1}(C)+W_{2}(C)+\cdots+W_{N}(C).$ (64)

Each Wilson loop $W_{i}$ in $W_{\mathrm{fd}}$ has the same string tension, and
for large loops $C$ it obeys the area law:

$\Bigl{\langle}W_{\mathrm{fd}}(C)\Bigr{\rangle}\sim\exp(-T_{\bm{\mu}_{1}}\mathrm{Area}).$
(65)

Under the $1$-form symmetry,

$\mathbb{Z}_{N}^{[1]}:W_{\mathrm{fd}}\mapsto\mathrm{e}^{2\pi\mathrm{i}/N}W_{\mathrm{fd}},$
(66)

or, more precisely,

$\displaystyle\langle
U(C_{1})W_{\mathrm{fd}}(C_{2})\rangle=\exp\left({2\pi\mathrm{i}\over
N}\mathrm{Link}(C_{1},C_{2})\right)\langle W_{\mathrm{fd}}(C_{2})\rangle.$
(67)

In order to determine whether string tensions are controlled by the $1$-form
symmetry, let us consider the adjoint Wilson loop,

$W_{\mathrm{ad}}(C)=\sum_{i\not=j}W_{i}(C)W_{j}^{*}(C)=|W_{\mathrm{fd}}|^{2}-N.$
(68)

This has trivial $N$-ality, but we can readily check that its string tension
is _not_ zero using the result of Section 2.3:

$\Bigl{\langle}W_{\mathrm{ad}}(C)\Bigr{\rangle}\sim\exp(-T_{\bm{\alpha}}\mathrm{Area}),$
(69)

with $T_{\bm{\alpha}}\simeq 2T_{\bm{\mu}_{1}}$. This example clearly tells us
that the string tensions of the semi-Abelian gauge theory carry detailed data
of its gauge-group representations, which cannot be captured by the
$\mathbb{Z}_{N}^{[1]}$ symmetry.

Something new is needed to explain the failure of the $N$-ality rule, and this
is where the non-invertible topological lines Bhardwaj:2017xup ;
Buican:2017rxc ; Freed:2018cec ; Chang:2018iay ; Thorngren:2019iar ;
Ji:2019jhk ; Rudelius:2020orz ; Komargodski:2020mxz ; Aasen:2020jwb  come in.
We can easily construct such an operator by summing over all the
$\mathrm{S}_{N}$ conjugates of $U^{(1)}_{\theta}(C)$:

$\displaystyle\mathcal{T}_{\theta}(C)$ $\displaystyle\equiv$
$\displaystyle{1\over N!}\sum_{P\in\mathrm{S}_{N}}PU_{\theta}^{(1)}(C)P^{-1}$
(70) $\displaystyle=$ $\displaystyle{1\over
N(N-1)}\sum_{\bm{\alpha}\in\Phi}\exp\left(\mathrm{i}{\theta\over
2\pi}\int_{C}\bm{\alpha}\cdot\mathrm{d}\bm{\sigma}\right).$

Since this operator is $\mathrm{S}_{N}$ singlet, it can be a physical operator
of the $\mathrm{S}_{N}$-gauged theory. Since each operator in the sum is
topological, so too is $\mathcal{T}_{\theta}(C)$. Therefore, this
$\mathrm{S}_{N}$-invariant operator shares important features with the
$1$-form symmetry generators.666 Following the same logic, we can in fact
construct many more continuous families of non-invertible symmetries. Indeed,
we can average over all $\mathrm{S}_{N}$ conjugates of an arbitrary product
$U_{\theta_{1}}^{(k_{1})}(C)\cdots U_{\theta_{r}}^{(k_{r})}(C)$ of the
operators (60) to get a non-invertible symmetry generator
$\mathcal{T}_{\theta_{1},\ldots,\theta_{r}}^{(k_{1},\ldots,k_{r})}(C)$. In
particular, the non-invertible symmetry generator
$\mathcal{T}_{\theta_{1},\ldots,\theta_{N-1}}^{(1,\ldots,N-1)}(C)$ with
$\theta_{k}=2\pi k/N$ will coincide with the $\mathbb{Z}_{N}^{[1]}$ center
symmetry generator $U(C)$, as one can readily check from (62). Thus, the
$\mathbb{Z}_{N}^{[1]}$ center symmetry is actually contained within a
continuous family of non-invertible symmetries.  However, the group
multiplication law is not satisfied for $\mathcal{T}_{\theta}(C)$, as one can
easily check:

$\mathcal{T}_{\theta}(C)\mathcal{T}_{\theta^{\prime}}(C)\not=\mathcal{T}_{\theta+\theta^{\prime}}(C).$
(71)

Because of the violation of the group multiplication property, we cannot
regard $\mathcal{T}_{\theta}(C)$ as a generator of an ordinary $1$-form
symmetry in contrast with (62). Instead, it is a generator of a non-invertible
symmetry.

Let us consider the component of the Wilson loop that corresponds to the
weight $\bm{w}\in\Gamma_{\mathrm{w}}$. Its eigenvalue for
$\mathcal{T}_{\theta}$ is given by

${1\over
N(N-1)}\sum_{\bm{\alpha}\in\Phi}\exp\left(\mathrm{i}\theta\bm{\alpha}\cdot\bm{w}\right).$
(72)

As a consequence, the fundamental Wilson loop transforms as

$W_{\mathrm{fd}}\mapsto{1\over
N}\bigl{(}N-2+2\cos(\theta)\bigr{)}W_{\mathrm{fd}}.$ (73)

More importantly, the adjoint Wilson loop also transforms nontrivially as

$W_{\mathrm{ad}}\mapsto{(N-2)(N-3)+4(N-2)\cos(\theta)+2\cos(2\theta)\over
N(N-1)}W_{\mathrm{ad}}.$ (74)

This elucidates that we can detect the detailed information of the Wilson loop
beyond $N$-ality by using the non-invertible topological line operator
$\mathcal{T}_{\theta}$.

As another example, we can detect the difference between the symmetric and
anti-symmetric two-index representations, $W_{\mathrm{sym}}$ and
$W_{\mathrm{asym}}$, of $\mathrm{SU}(N)$, whose highest weights are given by
$2\bm{\mu}_{1}$ and $\bm{\mu}_{2}$, respectively. We have to note, however,
that $W_{\mathrm{sym}}$ is not an eigen-operator of $\mathcal{T}_{\theta}$,
because the two-index symmetric representation of $\mathrm{SU}(N)$ decomposes
into two irreducible representations of
$\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}$. Since
$\bm{\mu}_{2}=2\bm{\mu}_{1}-\bm{\alpha}_{1}$ and
$(2\bm{\mu}_{1})\cdot\bm{\alpha}_{1}=2$, each charge in the anti-symmetric
representation appears in the list of charges of the symmetric representation
exactly once, and thus the correct eigen-operator is
$W_{\mathrm{sym}}-W_{\mathrm{asym}}$. Indeed, one can check that

$\Bigl{\langle}(W_{\mathrm{sym}}-W_{\mathrm{asym}})(C)\Bigr{\rangle}\sim\exp(-2T_{\bm{\mu}_{1}}\mathrm{Area}),\qquad\Bigl{\langle}W_{\mathrm{asym}}(C))\Bigr{\rangle}\sim\exp(-T_{\bm{\mu}_{2}}\mathrm{Area}),$
(75)

with $T_{\bm{\mu}_{2}}<2T_{\bm{\mu}_{1}}$, as we have discussed in Section
2.3. We find

$(W_{\mathrm{sym}}-W_{\mathrm{asym}})\mapsto{N-2+2\cos(2\theta)\over
N}(W_{\mathrm{sym}}-W_{\mathrm{asym}}),$ (76)

and

$W_{\mathrm{asym}}\mapsto{(N-2)(N-3)+2+4(N-2)\cos(\theta)\over
N(N-1)}W_{\mathrm{asym}}.$ (77)

This gives another explicit demonstration of the fact that one can distinguish
different string tensions for representations of the same $N$-ality with the
help of the topological operator $\mathcal{T}_{\theta}$.

###  3.4 Effect of dynamical electric particles

In the previous section, we discussed the behavior of string tensions for the
pure semi-Abelian gauge theory. String tensions do not obey the $N$-ality
rule, and the presence of non-invertible topological lines explain why they
carry more detailed information. In this section, we discuss what will happen
to the string tensions once dynamical electric charges are added.

Once electric charges are incorporated as dynamical excitations, their pair
creation can break confining strings if it is energetically favorable. If the
fundamental electric charge is added, we expect that all the confining strings
can be broken and all Wilson loops will obey the perimeter law. If the adjoint
charge is added instead, we expect that the string tensions should obey the
$N$-ality rule, because the adjoint Wilson loop would then obey the perimeter
law. Can we justify these expectations from the viewpoint of topological
lines?

For this purpose, we need to identify which line operators cease to be
topological once the dynamical electric charges are included. If a line acts
nontrivially on the Wilson loop corresponding to the dynamical excitations,
then it is no longer topological after introducing dynamical charges
Rudelius:2020orz . This is because the corresponding Wilson loop can end on
charged local operators, so that the linking number is no longer well-defined;
in other words, the topological invariance of the symmetry operator is lost.

Let us add dynamical adjoint particles, and then determine whether or not
$\mathcal{T}_{\theta}$ is topological. Since the eigenvalue of
$W_{\mathrm{adj}}$ has to be $1$ for any topological operator,
$\mathcal{T}_{\theta}$ is topological only if

${(N-2)(N-3)+4(N-2)\cos(\theta)+2\cos(2\theta)\over N(N-1)}=1.$ (78)

This is solved only by $\theta=0$ mod $2\pi$, and thus only the trivial one
$\mathcal{T}_{\theta=0}=1$ is topological. This implies that the non-
invertible symmetry ceases to be an exact symmetry once an adjoint matter
field is added. On the other hand, the $\mathbb{Z}_{N}$ $1$-form symmetry is
kept intact because the generator acts trivially on $W_{\mathrm{ad}}$. In this
case, the string tensions obey the $N$-ality rule at least if the Wilson loops
are sufficiently large, which is consistent with the observation for $3$d
$\mathrm{SU}(N)$ Yang–Mills theory.

As a nontrivial exercise, we can add dynamical particles corresponding to
$n\Gamma_{\mathrm{r}}$ with $n>1$, instead of $W_{\mathrm{ad}}$. Then the non-
invertible line $\mathcal{T}_{\theta}$ is topological if

${(N-2)(N-3)+4(N-2)\cos(n\theta)+2\cos(2n\theta)\over N(N-1)}=1,$ (79)

and this has nontrivial solutions, $\theta\in(2\pi/n)\mathbb{Z}$. Therefore,
the continuous part of the non-invertible symmetry $\mathcal{T}_{\theta}$ is
explicitly broken by dynamical electric charges $n\Gamma_{\mathrm{r}}$, but
the discrete part $\mathcal{T}_{\theta=2\pi k/n},\;k=1,\ldots,n$ still
generates a good non-invertible symmetry. As a result, Wilson lines
distinguished by $\mathcal{T}_{\theta=2\pi k/n}$ can have different string
tensions even if they share the same $N$-ality.

##  4 Summary and Discussions

In this paper, we have studied the properties of the semi-Abelian gauge theory
in $3$ spacetime dimensions, where the gauge group is
$G_{\mathrm{gauge}}=\mathrm{U}(1)^{N-1}\rtimes\mathrm{S}_{N}$. As we have
imposed the flatness condition on the $\mathrm{S}_{N}$ gauge field, we can
locally eliminate it completely, so the spectral properties of the mass gap
and string tensions can be calculated as the $\mathrm{U}(1)^{N-1}$ theory. We
have seen that the mass gap is generated via the Polyakov mechanism as a
consequence of monopole-instanton proliferation. We can classify their
magnetic charges using the $\mathrm{SU}(N)$ representation, and all the
monopoles for the roots give equally dominant contributions to the effective
potential. This point is very different from the Polyakov model or QCD(adj)
with an $S^{1}$ compactification, where only the monopoles associated with the
simple roots play the dominant role, and it comes from the $\mathrm{S}_{N}$
invariance of our model.

Using the dual formulation, we also computed various string tensions, and we
found that there are infinitely many different string tensions. When the
$\mathrm{S}_{N}$ symmetry is not gauged, this can be explained very naturally
in the context of the $1$-form symmetry because the center of
$\mathrm{U}(1)^{N-1}$ is $\mathrm{U}(1)^{N-1}$ itself, and thus the $1$-form
symmetry group is large enough to explain the selection rules between
infinitely many confining strings.

A puzzle arises, however, after gauging $\mathrm{S}_{N}$, because the center
symmetry is just $Z(G_{\mathrm{gauge}})=\mathbb{Z}_{N}$. This is because most
of the elements of $\mathrm{U}(1)^{N-1}$ do not commute with the permutations,
and the permutation invariance requires that the center elements be
proportional to the identity matrix. Thus, the $1$-form symmetry of semi-
Abelian gauge theory is as small as that of $\mathrm{SU}(N)$ Yang–Mills
theory, where the string tensions are characterized by $N$-ality alone.
Therefore, for the semi-Abelian theory, there is a clear discrepancy between
the actual behavior of the string tensions and the natural expectation from
$\mathbb{Z}_{N}$ center symmetry.

We find that the discrepancy is resolved by recognizing the presence of
noninvertible symmetry. We constructed the topological line operator
$\mathcal{T}_{\theta}$ out of the $\mathrm{U}(1)^{N-1}$ $1$-form symmetry
generators, which remain well-defined and topological after gauging
$\mathrm{S}_{N}$ but do not satisfy the group multiplication law. Though this
operator is noninvertible, its action on the Wilson lines yield eigenvalues
that are able to distinguish representations with the same $N$-ality. Thus, we
have demonstrated the utility of an extended notion of symmetry in a $3$d toy
example of a gauge theory.

We should mention that the formal development of non-invertible symmetry is
still an important task. In the case of higher-form or higher-group symmetry,
their formalization not only provided the rigorous definition and
generalization of the center symmetry, but also gave new tools to analyze
interacting QFTs, such as generalizations of anomaly matching Gaiotto:2017yup
; Tanizaki:2017bam ; Komargodski:2017smk ; Shimizu:2017asf ; Gaiotto:2017tne ;
Tanizaki:2017qhf ; Tanizaki:2017mtm ; Tanizaki:2018wtg ; Komargodski:2017dmc ;
Kikuchi:2017pcp ; Tanizaki:2018xto ; Karasik:2019bxn ; Cordova:2019jnf . It
would be very nice if this repertoire of useful tools could be enhanced to
include non-invertible symmetry.

Lastly, let us present some speculation. As we stated in the introduction, a
similar behavior regarding the $N$-ality rule has been observed in simulations
of $\mathrm{SU}(N)$ Yang–Mills on the lattice: there is an intermediate
distance scale where the quark-antiquark potential exhibits linear confinement
but its string tension depends on the details of the gauge-group
representation. Though it is widely believed that the string tension becomes
solely dictated by $N$-ality once the quark-antiquark separation becomes
sufficiently large, it is logically possible that ‘sufficiently large’ is
parametrically larger than the strong length scale $\Lambda^{-1}$ at which
confinement sets in. For instance, viewing $N$ as a parameter, it may very
well be that the $N$-ality rule sets in at a distance scale
$h(N)\Lambda^{-1}\gg\Lambda^{-1}$, where $h(N)\rightarrow\infty$ as
$N\rightarrow\infty$. We think it would be an intriguing possibility if, even
in pure Yang–Mills, some approximate notion of non-invertible symmetry could
be used to explain the behavior of string tensions beyond $N$-ality at these
intermediate distances.

A more striking example may be QCD with fundamental or two-index matter
fields, where the 1-form ${\mathbb{Z}}_{N}$ center symmetry is either
completely or partially lost, or Yang–Mills theories with simply-connected
gauge groups without a center, such as $G_{2}$. Even in cases where 1-form
symmetry is completely lost, we believe that an approximate non-invertible
symmetry could potentially give a precise meaning to confinement of arbitrary
test charges, and therefore provide the long sought definition of confinement
in such theories.

###### Acknowledgements.

We thank Shailesh Chandrasekharan, Hanqing Liu, Hersh Singh, Misha Shifman,
Mike Creutz, Tin Sulejmanpasic, and Aleksey Cherman for useful discussions.
The authors thank the YITP–RIKEN iTHEMS workshop “Potential Toolkit to Attack
Nonperturbative Aspects of QFT –Resurgence and related topics–” (YITP-T-20-03)
for providing opportunities of useful discussions. The work of Y. T. was
partially supported by JSPS KAKENHI Grant-in-Aid for Research Activity Start-
up, 20K22350. M. Ü. is supported by the U.S. Department of Energy, Office of
Science, Division of Nuclear Physics under Award DE-SC0013036.

##  Appendix A Review of Abelian duality on the lattice

###  A.1 Differential forms on the lattice

There is, on the lattice, a close analog of the notion of differential forms,
and it is especially convenient for treating Abelian lattice gauge theories.
Here we give a somewhat informal introduction to this formalism, which we use
throughout this article. See Sulejmanpasic:2019ytl  for a more systematic
discussion.

We begin with a bit of lattice geometry. Consider a $d$-dimensional cubic
lattice $\Lambda^{d}$. Such a lattice contains ‘$r$-cells’ $c^{(r)}$ for each
$r=0,1,\ldots,d$. Thus, the $0$-cells are the sites $s$, the $1$-cells the
links $\ell$, the $2$-cells the plaquettes $p$, the $3$-cells the cubes $c$,
etc., and everything is oriented. For example, for a link
$\ell=(x;\widehat{{\mu}})$, the oppositely oriented link is given by
$-\ell\equiv(x+\widehat{{\mu}};-\widehat{{\mu}})$, and these are to be viewed
as distinct objects despite corresponding to the same unoriented edge.777In
the case of sites, an opportunity for confusion may arise. In this notation,
the sites $s$ and $-s$ correspond to the same point $x$, say, but are equipped
with opposite orientations. By convention, whenever we write a sum or product
over $r$-cells, we do not double count $r$-cells that only differ by
orientation.

The ‘boundary operator’ ${\partial}$ takes an $r$-cell into the (oriented) sum
of the $(r-1)$-cells that constitute its boundary. For example, the boundary
operator on a plaquette $p=(x;\widehat{{\mu}},\widehat{{\nu}})$ yields

${\partial}(x;\widehat{{\mu}},\widehat{{\nu}})=(x;\widehat{{\mu}})+(x+\widehat{{\mu}};\widehat{{\nu}})-(x+\widehat{{\nu}};\widehat{{\mu}})-(x;\widehat{{\nu}}).$
(80)

Importantly, the boundary operator is nilpotent, ${\partial}^{2}=0$. By a
slight abuse of notation, we write

$c^{(r-1)}\subset{\partial}c^{(r)}$ (81)

if the $r$-cell $c^{(r)}$ contains in its boundary the $(r-1)$-cell
$c^{(r-1)}$. We thus have (tautologically)

${\partial}c^{(r)}=\sum_{c^{(r-1)}\subset{\partial}c^{(r)}}c^{(r-1)}.$ (82)

We also have a kind of dual to the boundary operator, the ‘coboundary
operator’ $\delta$. It takes an $r$-cell into the sum of the $(r+1)$-cells
that each contains $c^{(r)}$ in its boundary,

$\delta c^{(r)}=\sum_{{\partial}c^{(r+1)}\supset c^{(r)}}c^{(r+1)}.$ (83)

For example, for a link $\ell=(x;\widehat{1})$ in a three-dimensional lattice,
we have

$\delta(x;\widehat{1})=(x;\widehat{1},\widehat{3})+(x;\widehat{1},\widehat{2})+(x-\widehat{3};\widehat{3},\widehat{1})+(x-\widehat{2};\widehat{2},\widehat{1})$
(84)

(see Figure 2).

Figure 2: Coboundary operator on a link in a 3d lattice:
$\delta\ell=p_{1}+p_{2}+p_{3}+p_{4}$

It is easy to show that $\delta^{2}=0$.

There is another lattice $\tilde{{\Lambda}}^{d}$, the ‘dual lattice’, that is
naturally associated with the primary lattice ${\Lambda}^{d}$. The points of
$\tilde{{\Lambda}}^{d}$ are given by
$\tilde{x}=x+\frac{1}{2}(\widehat{1}+\cdots+\widehat{d})$, with $x$ any point
of ${\Lambda}^{d}$. These lattices are connected by an operator $*$, which
takes $r$-cells in the primary lattice into $(d-r)$-cells in the dual lattice
and vice versa; it is defined as follows: for an $r$-cell $c^{(r)}$ in the
primary lattice, $*c^{(r)}$ is the unique $(d-r)$-cell in the dual lattice
such that $c^{(r)}$ and $*c^{(r)}$ intersect transversally, and such that the
orientation of the ordered pair $(c^{(r)},*c^{(r)})$ is positive. For example,
for a plaquette $p=(x;\widehat{1},\widehat{2})$ in a 3d lattice
${\Lambda}^{3}$, we have $*p=(\tilde{x}-\widehat{3};\widehat{3})$ (see Figure
1). On $r$-cells, we have

$*^{2}=(-)^{r(d-r)}.$ (85)

We can now define differential forms on the lattice. An $r$-form ${\omega}$ is
simply a gadget that assigns a value ${\omega}_{c^{(r)}}$ to each $r$-cell
$c^{(r)}$, and it extends as a linear map. To compare with more conventional
lattice field theory notation, consider for example a 1-form ${\theta}$. We
may write its value on a link $\ell=(x;\widehat{{\mu}})$ as

${\theta}_{\ell}\equiv{\theta}_{\mu}(x).$ (86)

We define the ‘exterior differential’ operator d to take $r$-forms to
$(r+1)$-forms according to the formula

$(\text{d}{\omega})_{c^{(r+1)}}\equiv\sum_{c^{(r)}\subset{\partial}c^{(r+1)}}{\omega}_{c^{(r)}}={\omega}_{{\partial}c^{(r+1)}}.$
(87)

To again compare with more conventional notation, we note that the
differential $\text{d}{\theta}$ of the 1-form ${\theta}$ on a plaquette
$p=(x;\widehat{{\mu}},\widehat{{\nu}})$ is given by

$(\text{d}{\theta})_{p}=\theta_{\mu}({x})+\theta_{\nu}({x+\widehat{{\mu}}})-\theta_{\mu}({x+\widehat{{\nu}}})-\theta_{\nu}({x}).$
(88)

We also define the dual $\text{d}^{\dagger}$ of the exterior differential, the
‘codifferential’, which takes $r$-forms to $(r-1)$-forms, according to the
formula

$(\text{d}^{\dagger}{\omega})_{c^{(r-1)}}\equiv\sum_{c^{(r)}\subset\delta
c^{(r-1)}}{\omega}_{c^{(r)}}={\omega}_{\delta c^{(r-1)}}.$ (89)

It is easy to see that $\text{d}^{2}=(\text{d}^{\dagger})^{2}=0$. The star
operator $*$ takes $r$-forms on the primary lattice to $(d-r)$-forms on the
dual lattice, and vice versa, according to the formulae

$(*{\omega})_{\tilde{c}^{(d-r)}}={\omega}_{*\tilde{c}^{(d-r)}}\,,\qquad(*\tilde{{\omega}})_{c^{(d-r)}}=\tilde{{\omega}}_{*c^{(d-r)}}.$
(90)

It is easy to show that on $r$-forms, we have

$*^{2}=(-)^{r(d-r)}.$ (91)

One of the more useful features of lattice form notation is that it enables us
to ‘integrate by parts’ mindlessly. That is, we have the formula

$\sum_{c^{(r)}}(\text{d}{\omega})_{c^{(r)}}{\tau}_{c^{(r)}}=\sum_{c^{(r-1)}}{\omega}_{c^{(r-1)}}(\text{d}^{\dagger}{\tau})_{c^{(r-1)}}.$
(92)

Actually, it is this partial integration formula that justifies calling
$\text{d}^{\dagger}$ the dual of d. To illustrate the utility of the notation,
let us prove (92):

$\mathrm{L.H.S.}\equiv\sum_{c^{(r)}}\sum_{c^{(r-1)}\subset{\partial}c^{(r)}}{\omega}_{c^{(r-1)}}{\tau}_{c^{(r)}}=\sum_{c^{(r-1)}}\sum_{c^{(r)}\subset\delta
c^{(r-1)}}{\omega}_{c^{(r-1)}}{\tau}_{c^{(r)}}\equiv\mathrm{R.H.S}.$ (93)

Finally, let us discuss the lattice analog of the ‘Hodge decomposition’. As in
the continuum, we define the Laplacian on forms by
${\Delta}=\text{d}\text{d}^{\dagger}+\text{d}^{\dagger}\text{d}$. In
particular, on 0-forms ${\varphi}$, we have

$(\Delta{\varphi})(x)=\sum_{{\mu}=1}^{d}\large[2{\varphi}(x)-{\varphi}(x+\widehat{{\mu}})-{\varphi}(x-\widehat{{\mu}})\large].$
(94)

Forms annihilated by ${\Delta}$ are called ‘harmonic’. It is simple to show
that harmonic forms are annihilated by both d and $\text{d}^{\dagger}$. The
Hodge decomposition is the statement that any $r$-form ${\omega}$ can be
written uniquely as

${\omega}=\text{d}{\alpha}+\text{d}^{\dagger}{\beta}+{\eta},$ (95)

where ${\eta}$ is harmonic. We will not prove this here.

###  A.2 3d compact QED on the lattice

Here we discuss the dual representation of 3d $\mathrm{U}(1)$ lattice gauge
theory following the presentation of Ref. Gopfert:1981er  (see also
Banks:1977cc ; Savit:1977fw ). We also give some attention to global issues
involving the spacetime topology. We note that, although we restrict our
presentation to three dimensions, many techniques used here are also
applicable in four-dimensional spacetime lattices, where interesting phase
diagrams have been expected through electromagnetic dualities Creutz:1979kf ;
Creutz:1979zg ; Cardy:1981qy ; Cardy:1981fd ; Honda:2020txe .

We start from the Wilson formulation of the $\mathrm{U}(1)$ lattice gauge
theory in $d=3$ spacetime dimensions:

$\displaystyle\exp(-S)$ $\displaystyle=\exp\left(\beta\sum_{p}(\cos
f_{p}-1)\right)=\prod_{p}\text{e}^{\beta(\cos f_{p}-1)}.$ (96)

Since the action is periodic in $f_{p}$, we can expand $\exp(-S)$ as a Fourier
series:

$\displaystyle\text{e}^{\beta(\cos
f_{p}-1)}=\sum_{k_{p}\in\mathbb{Z}}\text{e}^{\text{i}k_{p}f_{p}}I_{k_{p}}(\beta)\text{e}^{-{\beta}},$
(97)

where $I_{k_{p}}(\beta)$ is the modified Bessel function of the first kind of
order $k_{p}$. This representation is useful because it allows us to integrate
over the link fields in the Abelianized theory in a straightforward manner.

The partition function can be rewritten as

$Z=\int_{0}^{2\pi}[\text{d}a_{\ell}]\exp(-S)=\sum_{\\{k_{p}\in\mathbb{Z}\\}}\int_{0}^{2\pi}[\text{d}a_{\ell}]\exp\left(\text{i}\sum_{p}k_{p}f_{p}\right)\prod_{p}I_{k_{p}}(\beta)\text{e}^{-{\beta}}.$
(98)

From this expression, in the weak coupling limit $\beta\gg 1$, we can obtain
the Villain form. Using the asymptotic expansion,
$\text{e}^{-\beta}I_{k_{p}}(\beta)\sim\frac{1}{\sqrt{2\pi\beta}}\text{e}^{-\frac{1}{2\beta}k_{p}^{2}}$,
we can rewrite the summation over $k_{p}$ by Poisson summation formula,

$\displaystyle\sum_{k_{p}\in\mathbb{Z}}\text{e}^{\text{i}k_{p}f_{p}}\frac{1}{\sqrt{2\pi\beta}}e^{-\frac{1}{2\beta}k_{p}^{2}}=\sum_{n_{p}\in\mathbb{Z}}\text{e}^{-\frac{\beta}{2}(f_{p}-2\pi
n_{p})^{2}}$ (99)

Here, $n_{p}$ can be viewed as the flux passing through the corresponding
surface $p$. The total flux passing through the surface of the cube $c$
centered at $\tilde{x}$ is

$\displaystyle\oint_{\partial c}n=q(\tilde{x}),$ (100)

which is just the magnetic charge located at $\tilde{x}$. In the following, we
concentrate only on this weak-coupling limit that is exactly equivalent to the
Villain formulation.

##### Dual formulation, from $\Lambda^{3}$ to $\tilde{\Lambda}^{3}$:

In order to obtain the dual representation of the Villain form, we perform the
exact integration over $a_{\ell}$ before the summation over $k_{p}$ in (98).
As
$\sum_{p}k_{p}(\mathrm{d}a)_{p}=\sum_{\ell}(\mathrm{d}^{\dagger}k)_{\ell}a_{\ell}$,
the exact integration over $a_{\ell}$ enforces the constraint,

$\displaystyle(\text{d}^{\dagger}k)_{\ell}=0.$ (101)

As a result, the partition function can be written as a constrained sum over
the $k_{p}$:

$\displaystyle
Z=\sum_{\\{k_{p}\in\mathbb{Z}\\}}\left\\{\prod_{\ell}\delta_{(\text{d}^{\dagger}k)_{\ell},0}\right\\}\left\\{\prod_{p}\text{e}^{-\frac{1}{2\beta}k_{p}^{2}}\right\\}.$
(102)

To construct the dual formulation of the theory, it is useful to turn the
constrained sum into an unconstrained sum. To this end, we consider the
decomposition of $(*k)_{\tilde{\ell}}$ as

$(*k)_{\tilde{\ell}}=(\mathrm{d}m)_{\tilde{\ell}}+\tilde{a}_{\tilde{\ell}},$
(103)

where $m_{\tilde{s}}$ is an integer-valued scalar field, and
$\tilde{a}_{\tilde{\ell}}$ is an integer-valued link field on the dual
lattice. Since $k_{p}$ satisfies the constraint (101), $\tilde{a}$ can be
regarded as a flat connection. In computing the partition function, we may
make the replacement, $k_{p}\rightarrow(*\text{d}m)_{p}+(*\tilde{a})_{p},$ and
the constrained sum over $\\{k_{p}\in\mathbb{Z}\\}$ becomes an unconstrained
sum over $\\{m_{\tilde{s}}\in\mathbb{Z}\\}$ and $[\tilde{a}]\in
H^{1}(\tilde{\Lambda},\mathbb{Z})$. As a result, the partition function in the
weak-coupling limit takes the simple form,

$Z=\sum_{[\tilde{a}]\in
H^{1}}\sum_{\\{m_{\tilde{s}}\in\mathbb{Z}\\}}\exp\left(-\frac{1}{2\beta}\sum_{\tilde{\ell}}(\text{d}m+\tilde{a})_{\tilde{\ell}}^{2}\right),$
(104)

after using the duality relation (103). This model is sometimes called the
${\mathbb{Z}}$-ferromagnet when $\tilde{a}=0$.

Two more steps are needed to convert ${\mathbb{Z}}$-ferromagnet representation
to a continuum QFT. First, convert the sum into an integration over a
continuous variable. Using the Poisson resummation identity repeatedly through
the lattice $\tilde{\Lambda}_{3}$,

$\displaystyle\sum_{m(\tilde{x})\in\mathbb{Z}}\delta(\sigma(\tilde{x})-2\pi
m(\tilde{x}))=\sum_{q(\tilde{x})\in\mathbb{Z}}\text{e}^{\text{i}\,q(\tilde{x})\sigma(\tilde{x})},$
(105)

we immediately obtain the partition function as an infinite dimensional
integral,

$\displaystyle Z$ $\displaystyle=\sum_{[\tilde{a}]\in
H^{1}}\int[\text{d}\sigma(\tilde{x})]\;\sum_{\\{q(\tilde{x})\\}}\;\exp\left(-\frac{1}{8\pi^{2}\beta}\sum_{\tilde{x}}(\partial_{\mu}^{-}\sigma({\tilde{x}})+2\pi\tilde{a}_{\mu}(\tilde{x}))^{2}+\text{i}\sum_{\tilde{x}}q({\tilde{x}})\sigma({\tilde{x}})\right),$
(106)

where $q({\tilde{x}})\in{\mathbb{Z}}$ has an interpretation as the magnetic
charge of a monopole-instanton at position
${\tilde{x}}\in\tilde{\Lambda}_{3}$. The kinetic term of this expression
clarifies that $\tilde{a}$ plays the role of the gauge field for the discrete
shift symmetry $\sigma\mapsto\sigma+2\pi$, and thus the dual photon field
$\sigma$ is $2\pi$-periodic scalar. Having made this point, for simplicity of
notation, we shall neglect the effect of nontrivial topology from now on, and
set $\tilde{a}=0$.

The Gaussian integration over $\sigma$ can be done exactly to produce the
Coulomb gas representation for the magnetic monopoles:

$\displaystyle Z$
$\displaystyle=\sum_{\\{q(\tilde{x})\\}}\exp{\left(\frac{1}{2\beta}\sum_{\tilde{x},\tilde{x}^{\prime}}(-4\pi^{2}\beta^{2})q({\tilde{x}})v(\tilde{x}-\tilde{x}^{\prime})q({\tilde{x}^{\prime}})\right)},$
(107)

where $v(\tilde{x}-\tilde{x}^{\prime})$ is the three dimensional Coulomb
interaction formulated on the lattice (lattice Green function), formally given
by $v(\tilde{x})=\Delta^{-1}$. Let us split this Green function into two parts
by adding and subtracting $(\Delta+M_{\mathrm{PV}}^{2})^{-1}$,

$\displaystyle\Delta^{-1}$
$\displaystyle=\Delta^{-1}-(\Delta+M_{\mathrm{PV}}^{2})^{-1}+(\Delta+M_{\mathrm{PV}}^{2})^{-1}$
(108)
$\displaystyle=\Delta^{-1}(1+\Delta/M_{\mathrm{PV}}^{2})^{-1}+(\Delta+M_{\mathrm{PV}}^{2})^{-1}$
(109)
$\displaystyle=u_{M_{\mathrm{PV}}}(\tilde{x})+w_{M_{\mathrm{PV}}}(\tilde{x}),$
(110)

where $u_{M_{\mathrm{PV}}}(\tilde{x})$ is the Green function of the Pauli-
Villars (PV) regulated Laplacian
$\Delta_{M_{\mathrm{PV}}}\equiv\Delta(1+\Delta/M_{\mathrm{PV}}^{2})$, and
$w_{M_{\mathrm{PV}}}(\tilde{x})$ is the Yukawa Green function.

With this decomposition, we reintroduce the scalar field $\sigma$ for the PV
regulated propagator $u_{M_{\mathrm{PV}}}(\tilde{x})$, and then we obtain the
partition function as

$Z=\int[\text{d}\sigma(\tilde{x})]\;\sum_{\\{q(\tilde{x})\\}}\;\text{e}^{\sum_{\tilde{x}}\left(-\frac{1}{8\pi^{2}\beta}\sigma({\tilde{x}})\Delta_{M_{\mathrm{PV}}}\sigma({\tilde{x}})+\text{i}\,q({\tilde{x}})\sigma({\tilde{x}})\right)-2\pi^{2}\beta\sum_{\tilde{x},\tilde{x}^{\prime}}q({\tilde{x}})w_{M_{\mathrm{PV}}}(\tilde{x}-\tilde{x}^{\prime})q({\tilde{x}^{\prime}})}.$
(111)

Since $w_{M_{\mathrm{PV}}}(\tilde{x}-\tilde{x}^{\prime})$ is a massive
propagator with the PV mass $M_{\mathrm{PV}}$, it is exponentially damping if
$|\tilde{x}-\tilde{x}^{\prime}|\gtrsim 1/M_{\mathrm{PV}}$. Therefore, we can
take
$w_{M_{\mathrm{PV}}}(\tilde{x}-\tilde{x}^{\prime})=w_{M_{\mathrm{PV}}}(0)\delta_{\tilde{x}-\tilde{x}^{\prime},0}$,
where $w_{M_{\mathrm{PV}}}(0)=v(0)-{\cal O}(1/M_{\mathrm{PV}})\approx
0.253-{\cal O}(1/M_{\mathrm{PV}})$. In this limit, the partition function
simplifies into

$Z=\int[\text{d}\sigma(\tilde{x})]\text{e}^{-\frac{1}{8\pi^{2}\beta}\sum_{\tilde{x}}\sigma({\tilde{x}})\Delta_{M_{\mathrm{PV}}}\sigma({\tilde{x}})}\sum_{\\{q(\tilde{x})\\}}\prod_{\tilde{x}}\mathrm{e}^{-2\pi^{2}\beta
v(0)(q({\tilde{x}}))^{2}+\text{i}q({\tilde{x}})\sigma({\tilde{x}})},$ (112)

In (112), $2\pi^{2}\beta v(0)(q({\tilde{x}}))^{2}$ has an interpretation as
the action of the configurations with magnetic charge $q({\tilde{x}})$. Let us
denote the minimal action by $S_{0}=2\pi^{2}\beta v(0)$.

We now perform the dilute-gas approximation as the leading-order semiclassical
approximation. We only take into account the minimal effect of the monopole-
instantons corresponding to $q(\tilde{x})=\pm 1$ seriously, and regard higher-
order effects in $\mathrm{e}^{-S_{0}}$ as unimportant. As a result, we may
approximate the sum over $q(\tilde{x})$ by

$\sum_{q(\tilde{x})}\mathrm{e}^{-S_{0}q(\tilde{x})^{2}+\text{i}q(\tilde{x})\sigma({\tilde{x}})}=\exp\left(2\mathrm{e}^{-S_{0}}\cos(\sigma({\tilde{x}}))\right)+\mathcal{O}(\mathrm{e}^{-2S_{0}}).$
(113)

Substituting this expression into (112), we obtain the local Lagrangian for
the dual photon field:

$Z=\
\int[\text{d}{\sigma}(\tilde{x})]\exp\left(-\frac{1}{8\pi^{2}\beta}\sum_{\tilde{x}}{\sigma}({\tilde{x}})\Delta_{M_{\mathrm{PV}}}{\sigma}({\tilde{x}})+2\mathrm{e}^{-S_{0}}\sum_{\tilde{x}}\cos({\sigma}(\tilde{x}))\right).$
(114)

##  Appendix B Wilson to Villain at weak coupling

As mentioned in Section 2.1, semi-Abelian ${\rm U}(1)^{N-1}$ gauge theory may
also be given in the Wilson formulation by taking the action

$S_{\rm W}={\beta}\sum_{p}\sum_{i=1}^{N}(1-\cos
f^{i}_{p})-\text{i}\sum_{\ell}\sum_{i=1}^{N}v_{\ell}a^{i}_{\ell},$ (115)

where the $a^{i}_{\ell}\in[0,2\pi]$ are ${\rm U}(1)$ gauge fields, the
$f^{i}_{p}=(\text{d}a^{i})_{p}$ are the corresponding field strengths, and
$v_{\ell}\in{\mathbb{Z}}$ is a Lagrange multiplier. This expression has
manifest ${\rm S}_{N}$ global symmetry. The purpose of this appendix is to
demonstrate the weak-coupling equivalence of this formulation and the Villain
one (3).

The first step is to use (97) for $\cos(f_{p}^{i})$ for $i=1,\ldots,N$ with
the weak-coupling approximation, and then to apply (99) for $i=1,\ldots,N-1$.
We obtain a new action

$S_{1}=\frac{{\beta}}{2}\sum_{p}\sum_{i=1}^{N-1}(f^{i}_{p}+2\pi
n^{i}_{p})^{2}-\text{i}\sum_{\ell}\sum_{i=1}^{N-1}v_{\ell}a^{i}_{\ell}+\frac{1}{2{\beta}}\sum_{p}k_{p}^{2}-\text{i}\sum_{p}k_{p}f^{N}_{p}-\text{i}\sum_{\ell}v_{l}a^{N}_{\ell},$
(116)

where we have introduced integer-valued plaquette-fields $n^{i}_{p}$
($i=1,\ldots,N-1$) and $k_{p}$, over which we must perform the summation in
the partition function. Then exact integration over $a_{\ell}^{N}$ gives the
constraint

$v=-\mathrm{d}^{\dagger}k,$ (117)

and then the action becomes

$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\frac{{\beta}}{2}\sum_{p}\sum_{i=1}^{N-1}(f^{i}_{p}+2\pi
n^{i}_{p})^{2}+\text{i}\sum_{\ell}\sum_{i=1}^{N-1}(\text{d}^{\dagger}k)_{\ell}a^{i}_{\ell}+\frac{1}{2{\beta}}\sum_{p}k^{2}_{p}$
(118) $\displaystyle=$
$\displaystyle\frac{{\beta}}{2}\sum_{p}\sum_{i=1}^{N-1}(f^{i}_{p}+2\pi
n^{i}_{p})^{2}+\text{i}\sum_{\ell}\sum_{i=1}^{N-1}k_{p}f^{i}_{p}+\frac{1}{2{\beta}}\sum_{p}k^{2}_{p}.$

Applying the Poisson summation formula in terms of $k_{p}$, we get

$S_{3}=\frac{{\beta}}{2}\sum_{p}\sum_{i=1}^{N-1}(f^{i}_{p}+2\pi
n^{i}_{p})^{2}+{{\beta}\over 2}\sum_{p}\left(\sum_{i=1}^{N-1}f^{i}_{p}+2\pi
n_{p}\right)^{2},$ (119)

where a new integer-valued plaquette-field $n_{p}$ has taken the place of
$k_{p}$.

For convenience, let us rewrite this last action in the form

$S_{3}=\frac{{\beta}}{2}\sum_{i=1}^{N-1}(f^{i}_{p}+b^{i}_{p})^{2}+\frac{{\beta}}{2}\left(\sum_{i=1}^{N-1}f^{i}_{p}+b_{p}\right)^{2}$
(120)

by defining $b^{i}_{p}\equiv 2\pi n^{i}_{p}$, $b_{p}\equiv 2\pi n_{p}$.
Completing the square then yields

$S_{3}=\frac{{\beta}}{2}\sum_{p}\sum_{i,j=1}^{N-1}D^{ij}(f^{i}_{p}+b^{i}_{p}+w_{p}/N)(f^{j}_{p}+b^{j}_{p}+w_{p}/N)+\frac{{\beta}}{2}\sum_{p}w_{p}^{2},$
(121)

where we have defined

$w_{p}\equiv b_{p}-\sum_{i=1}^{N-1}b^{i}_{p}\,,\qquad D^{ij}\equiv
1+\delta^{i,j}.$ (122)

At this point, we realize that if we are only interested in the weak coupling
regime, then since the fluctuations in $w_{p}$ are gapped and discrete, we are
entitled to set $w_{p}=0$. Making this step leaves us with the action

$S_{4}=\frac{{\beta}}{2}\sum_{p}\sum_{i,j=1}^{N-1}D^{ij}(f^{i}_{p}+b^{i}_{p})(f^{j}_{p}+b^{j}_{p}).$
(123)

Now we note that the unimodular matrix

$M^{ij}=\delta^{i,j}-\delta^{i+1,j}$ (124)

satisfies

$M^{\rm t}DM=C,$ (125)

where $C$ is the Cartan matrix of ${\rm SU}(N)$:

$C^{ij}=\bm{\alpha}^{i}\cdot\bm{\alpha}^{j}=2\delta^{i,j}-\delta^{i,j+1}-\delta^{i+1,j}.$
(126)

It follows that we can make the field redefinitions

$a^{i}_{\ell}\rightarrow\sum_{j=1}^{N-1}M^{ij}A^{j}_{\ell}\,,\qquad
b^{i}_{p}\rightarrow\sum_{j=1}^{N-1}M^{ij}B^{j}_{p}$ (127)

with $A^{i}_{\ell}\in[0,2\pi]$, $B^{i}_{p}\in 2\pi{\mathbb{Z}}$, by the
unimodularity of $M$. This yields

$S=\frac{{\beta}}{2}\sum_{p}\sum_{i,j=1}^{N-1}C^{ij}(F^{i}_{p}+B^{i}_{p})(F^{j}_{p}+B^{j}_{p})$
(128)

which is equivalent to (3).

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