11institutetext: Department of Physics, North Carolina State University,
Raleigh, NC 27607, USA

# The metamorphosis of semi-classical mechanisms of confinement: From
monopoles on ${\mathbb{R}}^{3}\times S^{1}$ to center-vortices on
${\mathbb{R}}^{2}\times T^{2}$

Canberk Güvendik 1 Thomas Schaefer 1 Mithat Ünsal

###### Abstract

There are two distinct regimes of Yang-Mills theory where we can demonstrate
confinement, the existence of a mass gap, and fractional theta angle
dependence using a reliable semi-classical calculation. The two regimes are
Yang-Mills theory on $S^{1}\times{\mathbb{R}}^{3}$ with a small circle and a
double-trace deformation, and Yang-Mills theory on
$T^{2}\times{\mathbb{R}}^{2}$ where the torus $T^{2}$ is small and threaded by
a ’t Hooft flux. In the first case the confinement mechanism is related to
self-dual monopoles, whereas in the second case self-dual center-vortices play
a crucial role. These two topological objects are distinct. In particular,
they have different mutual statistics with Wilson loops. On the other hand,
they carry the same topological charge and action. On ${\mathbb{R}\times
T^{2}\times S^{1}}$, we are able to extrapolate both monopole regime and
vortex regime to a quantum mechanical domain, where a cross-over takes place.
Both sides of the cross-over are described by a deformed ${\mathbb{Z}}_{N}$
TQFT. On ${\mathbb{R}^{2}\times S^{1}\times S^{1}}$, we derive the effective
field theory of vortices from the effective theory of monopoles in the
presence of a ’t Hooft flux. This results from a two-stage adjoint Higgs
mechanism, to $U(1)^{N-1}$ in 3d first and a ${\mathbb{Z}}_{N}$ EFT in 2d
second. This proves adiabatic continuity of the two confinement mechanisms
across dimensions and shows how monopoles and their magnetic flux transmute
into center-vortices. This basic mechanism is flux fractionalization: The
magnetic flux of the monopoles fractionalizes and collimates in such a way
that 2d Wilson loops detect it as a center vortex.

††preprint: .

##  1 Introduction

Pure Yang-Mills theory and QCD(adj) with massless or massive fermions in the
adjoint representation are theories in which there exists a
${\mathbb{Z}}_{N}^{[1]}$ 1-form symmetry, or center-symmetry, and the notion
of confinement is sharply characterized in terms of the Wilson loop as an
order parameter. Both of these theories admit a semi-classical weak coupling
description for (non-thermal) compactifications of the theory on four-
manifolds such as ${\mathbb{R}}^{3}\times S^{1}$ Unsal:2007vu ; Unsal:2007jx ;
Unsal:2008ch ; Shifman:2008ja ; Davies:2000nw ; Poppitz:2021cxe  and
${\mathbb{R}}^{2}\times T^{2}$ Tanizaki:2022ngt ; Tanizaki:2022plm ;
Hayashi:2023wwi ; Hayashi:2024qkm . These realizations are adiabatically
connected to the theory on ${\mathbb{R}}^{4}$. In the limit where the size of
the circle or the torus are small non-perturbative effects such as
confinement, dependence on the theta angle, and the mass gap can be understood
analytically.

An important challenge that arises from these studies pointed in
Tanizaki:2022ngt  is the following: On ${\mathbb{R}}^{3}\times S^{1}$, where
the theory is guaranteed to remain center-symmetric form small $S^{1}$ via
either judicious boundary conditions for fermions or double-trace
deformations, confinement and other non-perturbative effects are generated by
the proliferation of monopole instantons or magnetic bions Unsal:2007vu ;
Unsal:2007jx ; Unsal:2008ch ; Shifman:2008ja ; Poppitz:2012sw ; Davies:2000nw
. On ${\mathbb{R}}^{2}\times T^{2}$ however, in the presence of a ’t Hooft
flux, it was shown that confinement is caused by the proliferation of center
vortices on ${\mathbb{R}}^{2}$ in a manner reminiscent of the charge-$N$
abelian Higgs model in 2d Tanizaki:2022ngt ; Tanizaki:2022plm ;
Hayashi:2023wwi ; Hayashi:2024qkm ; Hayashi:2024gxv . (See also
Poppitz:2022rxv ; Cox:2021vsa ; Anber:2023sjn ; Yamazaki:2017dra ;
Yamazaki:2017ulc .) Remarkably, both monopole instantons and center-vortices
can be viewed (in specified circumstances) as configurations with fractional
topological charge and fractional action

$\displaystyle Q_{T}=\frac{1}{N},\qquad
S_{0}=\frac{1}{N}S_{I}=\frac{8\pi^{2}}{g^{2}N},$ (1)

where $S_{I}$ is the 4d instanton action Kraan:1998pm ; Kraan:1998sn ;
Lee:1998bb ; Lee:1997vp . However, these are physically distinct
configurations, in particular, center-vortices possess a non-trivial mutual
statistics with the Wilson loop Gonzalez-Arroyo:1998hjb ; Montero:1999by ;
Montero:2000pb ; Gonzalez-Arroyo:2023kqv , while monopoles do not. More
specifically, on ${\mathbb{R}}^{3}\times S^{1}$, monopoles appear in
circumstances where Polyakov loop around $S^{1}$ acquires a non-trivial
profile $P_{3}\propto C$ where $C$ is clock matrix, leading to dynamical
abelianization, $SU(N)\rightarrow U(1)^{N-1}$. In contrast, on
${\mathbb{R}}^{2}\times T^{2}$, center-vortices appear when the Polyakov loops
on $T^{2}$ cycles are non-commuting pairs, $P_{1}=C,P_{2}=S$, such as clock
and shift matrix, and the gauge structure is reduced down to the center,
$SU(N)\rightarrow{\mathbb{Z}}_{N}$.

$\displaystyle{\mathbb{R}}^{3}\times S^{1}:\qquad
SU(N)\;\xrightarrow{P_{3}\;\text{adjoint Higgs }}\;U(1)^{N-1}\qquad{\rm
monopoles,\;bions,}$ (2) $\displaystyle{\mathbb{R}}^{2}\times T^{2}:\qquad
SU(N)\;\xrightarrow{P_{1,2}\;\text{adjoint
Higgs}}\;{\mathbb{Z}}_{N}\qquad\qquad{\rm center\;vortices}\,.$ (3)

Remarkably, despite the differences in the dynamics and mechanisms of
confinement, the qualitative and some quantitative features of the two regimes
are identical. This implies that these two _s_ emiclassical regimes may indeed
be _adiabatically connected_ , without any intervening phase transitions. In
this paper, we show how one regime transmutes to the other with the change of
parameters.

The long distance theory on ${\mathbb{R}}^{3}\times S^{1}$ is described as an
effective field theory (EFT) based on the grand canonical description of a
magnetic Coulomb gas Unsal:2008ch , similar to Polyakov model on
${\mathbb{R}}^{3}$ Polyakov:1975rs ; Polyakov:1987ez . We will refer to this
theory as the monopole EFT. In the long-distance regime on ${\mathbb{R}}\times
T^{2}\times S^{1}$, and ${\mathbb{R}}^{2}\times T^{2}$, in the appropriate
regimes, the long distance effective theory is a ${\mathbb{Z}}_{N}$
topological quantum field theory (TQFT) Banks:2010zn ; Kapustin:2014gua
deformed by local topological operators Nguyen:2024ikq ; Cherman:2021nox  in
1d and 2d, respectively as summarized in Fig.1. The local topological
operators are center-vortices in 2d and fractional instantons in 1d. We show
analytically the transition between these effective field theories:

$\displaystyle{\rm Monopole\;EFT\;in\;3d}\Longrightarrow{\rm
TQFT+deformation\;in\;1d\;and\;2d}$ (4)

as well as the relation between monopoles on ${\mathbb{R}}^{3}\times S^{1}$,
vortices on ${\mathbb{R}}^{2}\times T^{2}$ and fractional instantons on
${\mathbb{R}}\times T^{2}\times S^{1}$.

Figure 1: a) Regimes of $SU(N)$ gauge theory on ${\mathbb{R}}\times
T^{2}\times S^{1}$ with $n_{12}=1$. For ${\rm
min}(L_{S^{1}},L_{T^{2}})\ll\Lambda^{-1}$, the theory is amenable to weak
coupling semi-classical treatment and continuously connected to strong
coupling domain. Gauge structure is Higgsed to $U(1)^{N-1}$ in monopole
regime, and to ${\mathbb{Z}}_{N}$ in center-vortex regime. The two are
adiabatically connected, with a cross-over in between along the red line. See
§.3.3. b) Construction of Ref.Hayashi:2024yjc  in ${\mathbb{R}}^{2}\times
S^{1}\times S^{1}$ with $n_{23}=1$ units of ’t Hooft flux.

We provide a detailed discussion of monopole-vortex continuity in two
different settings. On ${\mathbb{R}}\times T^{2}\times S^{1}$ with a flux
through $T^{2}$ tHooft:1979rtg ; tHooft:1981sps ; vanBaal:1982ag , we describe
two regimes: one in which semiclassics is dictated by monopoles, and a
complementary regime in which vortices play crucial role. The two regimes meet
at a quantum mechanical domain, where a cross-over and dramatic rearrangement
of states occur, §.3.3.

On ${\mathbb{R}}^{2}\times T^{2}$, the most remarkable finding is the
transmutation of monopoles and their magnetic flux into center-vortices.111In
$d>2$ dimensions, most of the arguments in favor of center vortex mechanism
are based on numerical studies DelDebbio:1996lih ; DelDebbio:1998luz ;
Engelhardt:1998wu ; Alexandrou:1999iy ; Alexandrou:1999vx ; deForcrand:1999our
; deForcrand:2000pg ; Sale:2022qfn . In the center-vortex literature, there
are also models which view monopoles as junctions of center-vortices, see
e.g.Greensite:2003bk ; Deldar:2015kga ; Oxman:2018dzp . It is recently proven
that proliferation of center-vortices is a necessary, but not a sufficient
condition for confinement in $d>2$ Nguyen:2024ikq , while in $d=2$, it is also
sufficient. Our findings, in semi-classically calculable regimes of the Yang-
Mills theory may provide a basis for the mixed models. At short distances the
magnetic flux of monopole $2\pi\alpha_{i}$ is spherically symmetric. Within
the applicability of semi-classics and using a ’t Hooft flux tHooft:1979rtg ;
tHooft:1981sps ; vanBaal:1982ag , we show that the magnetic flux
fractionalizes and collimates into tubes of finite thickness. In the setting
of 2d EFT a large Wilson loop which encircles the core of the flux tube sees
it as center-vortex, their relation is topological and leads to well-know
phase factor $W(C)=e^{-\mathrm{i}\frac{2\pi}{N}}$ if vortex is inside $C$, and
$W(C)=1$ if it is outside $C$. In contrast, a Wilson loop measures the effect
of a monopole at a point $x\in{\mathbb{R}}^{3}$ through $\Omega_{D}(x)$, the
solid angle for the oriented surface $D$ $(\partial D=C)$ subtended at a point
$x$, and this is just geometrical. We provide a detailed explanation of this
phenomenon that we refer to as _flux fractionalization_ in §.4.1. We also
derive a 2d long-distance effective theory in terms of the degrees of freedom
of the 3d effective theory, the dual photons.

Also, starting with the classical long range Coulomb interaction between
monopoles, by compactifying in the presence of a ’t Hooft flux, we show that
the interaction remains long ranged in the ${\mathbb{R}}\times T^{2}\times
S^{1}$ with $n_{12}=1$, and the theory reduces to a multi-component Coulomb
gas in 1d quantum mechanical regime. While in the ${\mathbb{R}}^{2}\times
T^{2}$ with $n_{23}=1$, the interaction turns into the short-range classical
interactions between center-vortices in 2d EFT. Despite the fact that the
interaction between the center-vortices becomes short range, they still lead
to confinement due to the non-trivial mutual statistics with the Wilson loops.

These results are remarkable, because they provide a unified perspective of
monopole-induced confinement on ${\mathbb{R}}^{3}\times S^{1}$ and center-
vortex-induced confinement on ${\mathbb{R}}^{2}\times T^{2}$. From the
perspective of the grand-canonical ensemble of the two topological objects the
Wilson loop is disordered because of the sum over the induced phase factors.
Desctructive interference of these phases leads to the exponential smallness
of $\langle W(C)\rangle=e^{-{\rm Area}(D)T}$. Historically, the monopole and
center vortex pictures of confinement in QCD were viewed as distinct
alternatives for the confinement mechanism on ${\mathbb{R}}^{4}$
tHooft:1977nqb ; Cornwall:1979hz ; Nielsen:1979xu . However, it is recently
proven that center-vortex proliferation in $d\geq 3$ is a necessary, but not
sufficient condition for confinement Nguyen:2024ikq , whereas in $d=2$, it is
both necessary and sufficient. In the preent work we demonstrate that both
mechanisms are realized in different semi-classical regimes of the Yang-Mills
theory which are adiabatically connected to ${\mathbb{R}}^{4}$.

Perhaps more importantly, since both center-vortices and monopoles are
realized as fractional instantons in the appropriate semi-classical limits,
these results highlight the non-trivial inner structure of the 4d instanton.
It is by no means a simple structureless topological lump. We may hope that
these types of reliable semi-classical studies may reveal the true nature of
the 4d instanton, and provide a first principle study of non-perturbative
phenomena directly on ${\mathbb{R}}^{4}$.

Set-up: Throughout the paper, we study $4$-dimensional $SU(N)$ Yang-Mills (YM)
theory on $4$d spacetime $M_{4}$. The classical action of the theory is given
by

$\displaystyle S_{\mathrm{YM}}={1\over g^{2}}\int\mathrm{tr}[F(a)\wedge\star
F(a)]+{\mathrm{i}\,\theta\over 8\pi^{2}}\int\mathrm{tr}[F(a)\wedge F(a)]\,,$
(5)

where $a$ is the $SU(N)$ gauge field, $F(a)=\mathrm{d}a+\mathrm{i}a\wedge a$
is the field strength, $\theta$ is the topological theta angle, and ${1\over
8\pi^{2}}\int\mathrm{tr}[F(a)\wedge F(a)]\in{\mathbb{Z}}$ is the instanton
number. The theory has a $\mathbb{Z}_{N}$ $1$-form symmetry Gaiotto:2014kfa
(called center-symmetry in the older literature) denoted by
$\mathbb{Z}_{N}^{[1]}$. The 4-manifolds we consider are

$\displaystyle
M_{4}=\mathbb{R}\times(S^{1})_{1}\times(S^{1})_{2}\times(S^{1})_{3}$
$\displaystyle\quad{\rm with}\quad(x_{0},x_{1},x_{2},x_{3})\in M_{4},$ (6)

and circle sizes $L_{1},L_{2},L_{3}$, which includes the limits
${\mathbb{R}}^{3}\times S^{1}$ and ${\mathbb{R}}^{2}\times T^{2}$. In §.2 and
§.3, we turn on $n_{12}=1$ units of ’t Hooft flux, and work with a symmetric
$T^{2}$ with $L_{1}=L_{2}=L_{T^{2}}$. We explore different semi-classical
regimes by changing $L_{T^{2}}/L_{3}$. In §.4, we work with $n_{23}=1$ on
${\mathbb{R}}^{2}\times(S^{1})_{2}\times(S^{1})_{3}$ and an asymmetric torus,
and explore the dynamics by adjusting $L_{2}/L_{3}$.

Note added: During the completion of this work, we learned about
Ref.Hayashi:2024yjc , which uses $\mathbb{R}^{2}\times T^{2}$ with $n_{23}=1$.
Our construction in §.2 and §.3 uses $n_{12}=1$, and gives a complementary
perspective to monopole-vortex transmutation. Our Section. 4 and Appendix B
are motivated by this work. Different from Ref.Hayashi:2024yjc , we derive the
2d effective theory in terms of dual photon variables of the 3d monopole
theory.

##  2 Semi-classics on ${\mathbb{R}}\times T^{2}_{\rm large}\times S^{1}_{\rm
small}$ with ’t Hooft flux through $T^{2}$

We will describe various regimes of the adibatic continuity/cross-over diagram
Fig. 1 for Yang-Mills theory. We will review the necessary ingredients of the
semi-classical description on a small $S^{1}\times{\mathbb{R}}^{3}$
Unsal:2008ch ; Poppitz:2021cxe  and small $T^{2}\times{\mathbb{R}}^{2}$
Tanizaki:2022ngt  as we proceed. Our goal is to connect the semi-classical
theory in these two regimes through their common quantum mechanical limit in
Fig.1.

Consider the theory formulated on the 4-manifold:

$\displaystyle M_{4}$
$\displaystyle=\mathbb{R}\times\underbrace{(T^{2})_{12}}_{n_{12}=1\;{\rm
flux}}\times(S^{1})_{3}$ (7)

with the limiting cases

$\displaystyle{\mathbb{R}}\times T^{2}_{\rm large}\times S^{1}_{\rm
small}\sim{\mathbb{R}}^{3}\times S^{1},$ (8) $\displaystyle{\mathbb{R}}\times
T^{2}_{\rm small}\times S^{1}_{\rm large}\sim{\mathbb{R}}^{2}\times T^{2},$
(9)

for the large-$T^{2}$ and large-$S^{1}$ limits, respectively. Here, we assume

$\displaystyle{\rm min}(L_{S^{1}},L_{T^{2}})\ll\Lambda^{-1}\,,$ (10)

where $\Lambda^{-1}$ is the strong length scale of the Yang-Mills theory, and
this guarantees a weak coupling since $g^{2}N(L_{\rm short})\ll 1$. We expect
that both of these regimes are adiabatically connected to the
${\mathbb{R}}^{4}$ limit, as well as to each other since we can reproduce the
non-perturbative properties such as confinement, mass gap, and the multi-
branch structure of vacua. Since the theory is in the weak coupling regime
provided ${\rm min}(L_{S^{1}},L_{T^{2}})\ll\Lambda^{-1}$, it may be possible
to understand the ”transition” or ”cross-over” between these two confinement
regimes by using weak coupling tools. Our goal is to explore this connection
in this paper.

###  2.1 ’t Hooft flux: Alternative implementations and energetics

A ’t Hooft flux on $T^{2}$ is characterized by the gauge invariant integers
$n_{12}\in{\mathbb{Z}}_{N}$ that appear in the twisted boundary conditions.
The identification of fields at $(x_{1},x_{2})$, $(x_{1}+L_{1},x_{2})$ and
$(x_{1},x_{2}+L_{2})$ is achieved by using the transition functions
$g_{1}(x_{2}),g_{2}(x_{1})$.

$\displaystyle
a(x_{1}=L_{1},x_{2})=g_{1}(x_{2})^{\dagger}a(x_{1}=0,x_{2})g_{1}(x_{2})-\mathrm{i}g_{1}(x_{2})^{\dagger}\mathrm{d}g_{1}(x_{2}),$
(11) $\displaystyle
a(x_{1},x_{2}=L_{2})=g_{2}(x_{1})^{\dagger}a(x_{1},x_{2}=0)g_{2}(x_{1})-\mathrm{i}g_{2}(x_{1})^{\dagger}\mathrm{d}g_{2}(x_{1})\,.$
(12)

The consistency condition requires that tHooft:1979rtg

$g_{1}(L_{2})^{\dagger}g_{2}(0)^{\dagger}=g_{2}(L_{1})^{\dagger}g_{1}(0)^{\dagger}\exp\left(\mathrm{i}{\frac{2\pi}{N}}n_{12}\right).$
(13)

This label $n_{12}\in\mathbb{Z}_{N}$ is the ’t Hooft flux.

Turning on an $F_{12}$ or not: Although not often emphasized, the way
$n_{12}\neq 0$ is achieved in order to probe low energy physics of a given
theory is related to energetics Unsal:2020yeh . The condition $n_{12}\neq 0$
can always be achieved without turning on any chromo-magnetic field
$(F_{12}=0)$,222 We do not typically associate a chromomagnetic field with the
’t Hooft flux. However, it is possible to do so. In unbroken $SU(N)$ gauge
theories, this will also give an $n_{12}$ flux sector, but with higher energy.
However, if we know theory is abelianized (say by an adjoint Higgs mechanism),
then a lower energy configuration is often achieved by turning on $F_{12}\neq
0$. We thank Erich Poppitz for this explanation. but it can also be achieved
by turning on a non-vanishing abelian chromo-magnetic field $(F_{12}\neq 0)$
GarciaPerez:2013idu ; Unsal:2020yeh ; Cox:2021vsa ; Poppitz:2022rxv . In order
to probe the low-energy theory, which one of these should one use is a matter
of energetics and dynamics. In particular, in theories which are abelianized
by the Polyakov loop acting as an adjoint Higgs field, the case $(F_{12}\neq
0)$ may be favorable. We will encounter both in the following.

In compactification of the theories on ${\mathbb{R}}^{d}$ down to
${\mathbb{R}}^{d-1}\times S^{1}$, the holonomy of the gauge field in the
compact direction $P_{3}=e^{i\oint a_{3}}$ acts as a compact scalar adjoint
Higgs field from the perspective of the long distance theory. Depending on the
form of the $P_{3}$, this can lead to abelianization $SU(N)\rightarrow
U(1)^{N-1}$ or not. Thus, with further compactification of
${\mathbb{R}}^{d-1}\times S^{1}$ to eg. $\mathbb{R}\times T^{2}\times S^{1}$,
the dynamical behavior of Polyakov loop and the way ’t Hooft flux is
implemented ($F_{12}\stackrel{{\scriptstyle?}}{{=}}0)$ are not independent
questions.

Depending on the boundary conditions, we can have a Polyakov loop $P_{3}$
which is proportional to identity, or we can have $P_{3}$ proportional to the
clock matrix $C$. For example, on ${\mathbb{R}}^{3}\times S^{1}$, in the small
$S^{1}$ regime where $V(P_{3})$ is perturbatively calculable, we can have

$\displaystyle P_{3}^{\rm br.}$ $\displaystyle=\omega^{k}{\bf
1}_{N},\qquad\quad\omega=e^{\mathrm{i}\frac{2\pi}{N}},$ (14) $\displaystyle
P_{3}^{\rm ubr}$ $\displaystyle=\omega^{\alpha/2}C_{N}=\omega^{\alpha/2}{\rm
Diag}\left(1,\omega,\omega^{2},\ldots,\omega^{N-1}\right),\quad\alpha=0,1\;{\rm
for}\;N\in(2{\mathbb{Z}}+1,2{\mathbb{Z}}).$ (16)

The first one of these options takes place in pure thermal Yang-Mills at high-
temperature Gross:1980br , and the latter takes place in QCD(adj) with
massless or light fermions endowed with periodic boundary conditions
Kovtun:2007py . The latter can also be achieved by a double-trace deformation
Unsal:2008ch  or even with heavy adjoint fermions if $L_{3}$ is small
enoughCherman:2018mya ; Unsal:2010qh .

Assume we start with $L_{T^{2}}\rightarrow\infty$ as in (8), and consider the
center-unbroken configuration (16). Since we want to study Yang-Mills theory,
(16) can be accomplished either with double-trace deformation or by
introducing a heavy adjoint fermion with periodic boundary conditions, and
considering the small $L_{3}$ limit. A deformation potential which satisfies
center-stability on ${\mathbb{R}}\times T^{2}_{\rm large}\times S^{1}_{\rm
small}$ is333 The deformed theory, $S^{\rm dYM}=S^{\rm YM}+\Delta S$ satisfies
adiabatic continuity (absence of phase transitions) between the small
${\mathbb{R}}^{3}\times S^{1}$ and ${\mathbb{R}}^{4}$ theories. In the
large-$N$ limit, one can prove by using loop equations that the deformed small
${\mathbb{R}}^{3}\times S^{1}$ theory is equivalent to the theory on
${\mathbb{R}}^{4}$ theory Unsal:2008ch .

$\displaystyle\Delta S(P_{3})=-2\times S_{\rm 1-loop}^{\rm
YM}=2\times\int_{M_{4}}\sum_{n\geq 1}^{\lfloor
N/2\rfloor}\frac{2}{\pi^{2}L_{3}^{4}}\frac{1}{n^{4}}|\mathrm{tr}(P_{3})^{n}|^{2}\,$
(17)

where $S_{\rm 1-loop}^{\rm YM}$ is the one-loop effective action for the
Polyakov line in pure Yang-Mills theory Gross:1980br . Notice that deformation
is perturbative in ’t Hooft coupling $\lambda\equiv g^{2}N$ as it compensates
the one-loop effect, but unsuppressed in the large $N$ counting. It can be
produced dynamically at one loop by integrating out heavy adjoint fermions.
The deformation (17) ensures that the minimum of $V_{\rm 1-loop}^{\rm
YM}+\Delta V$ is located at (16), $P_{3}^{\rm ub}\propto C$.

Now, assume we turn on an $n_{12}=1$ ’t Hooft flux on $T^{2}_{\rm large}$. As
mentioned above, there are two possible scenarios: Constant non-abelian
transition matrices $g_{1}=C$, $g_{2}=S$ which is associated with $F_{12}=0$,
or abelian transition matrices leading to $F_{12}\neq 0$. These two options
have different energy costs. Energetically favorable one depends on the ratio
of $L_{3}$ and $L_{T^{2}}$.

Chromo-magnetic fluxes consistent with $n_{12}=1$ satisfy GarciaPerez:2013idu
444We thank A. Gonzalez-Arroyo and M. Garcia-Perez for discussions concerning
their work GarciaPerez:2013idu .

$\displaystyle F_{12}^{(i)}=\frac{2\pi}{L_{1}L_{2}}H^{(i)},\qquad H^{(i)}={\rm
Diag}(0,\ldots,0,1,0,\ldots,0)-\frac{1}{N}{\bm{1}}_{N}\,,$ (18)

where $H^{(i)}$ is the $i^{\rm th}$ Cartan generator. Such a configuration has
a classical energy, given by

$\displaystyle E_{\rm non-flat}=\frac{1}{2g^{2}}\int_{T^{2}\times
S^{1}}\mathrm{tr}\left[\left(F_{12}^{(i)}\right)^{2}\right]=\frac{L_{3}}{2L_{1}L_{2}}\left(\frac{2\pi}{g}\right)^{2}\left(1-\frac{1}{N}\right)\,.$
(19)

However, Yang-Mills theory with an abelian chromo-magnetic background field
has a tachyonic instability on ${\mathbb{R}}^{4}$, originally realized by
Nielsen-Olesen Nielsen:1978rm  for gluonic fields. Because of this, this
configuration was dismissed as a possible ground state in Ref.
GarciaPerez:2013idu . However, if the theory abelianizes by a potential for
the gauge holonomies, the tachyonic mode can and does disappear.555 For a
magnetic background field $B=B_{0}(\sigma_{3}/2)$ in $SU(2)$ gauge theory, the
mode decomposition of the field yields the frequencies
$\omega_{k_{z},n}^{2}=\Big{(}2B_{0}(n+\textstyle{\frac{1}{2}}+S_{3})+k_{z}^{2}\Big{)}$,
which is tachyonic for $S_{3}=-1$, $n=0$ where $k_{z}$ is continuous momenta
Nielsen:1978rm . However, on ${\mathbb{R}}\times T^{2}\times S^{1}$, with
$B_{0}=\frac{2\pi}{L_{1}L_{2}}$ and with non-trivial holonomy in the $S^{1}$
direction, this formula is modified to
$\omega_{k_{3},n}^{2}(\theta_{ij})=\Big{(}2B_{0}(n+\textstyle{\frac{1}{2}}+S_{3})+\frac{1}{L_{3}^{2}}(2\pi
k_{3}+(\theta_{i}-\theta_{j}))^{2}\Big{)}$ where in the center-symmetric
holonomy background, minimal value of $\theta_{ij}\sim\frac{2\pi}{N}$. Thus,
as long as center-symmetry is stable, for sufficiently large-
$(L_{T^{2}})^{2}$, the tachyonic instability disappears. This is the working
assumption in Unsal:2020yeh . However, as $(L_{T^{2}})$ gets smaller, at some
point, the tachyonic (Nielsen-Olesen) instability will kick in. This is
indicated by the red line in Fig. 1. This is however, not a phase transition,
as the theory is formulated in finite-volume. But there is a rather dramatic
rearrangement of states. We do not explore this cross-over in detail in this
paper. See §.3.3 for comparision of the two-sides of the crossover. It becomes
energetically favorable to work with such abelian backgrounds as argued in
Unsal:2020yeh . Notice that for large-$T^{2}$ with fixed ’t Hooft flux
$n_{12}=1$, the chromo-magnetic field dilutes as $1/{\rm Area}(T^{2})$, and
the classical total energy tends to zero asymptotically. Hence, as long as
$T^{2}$ is large enough, the ground states robustly lives in these chromo-
magnetic flux sectors.

In the presence of the ’t Hooft flux with non-abelian transition matrices and
$F_{12}=0$, the Polyakov loops around the nontrivial cycles of $T^{2}$ are
given by

$\displaystyle P_{1}(x_{2})$
$\displaystyle=g_{1}(x_{2})\,\mathcal{P}\mathrm{e}^{\mathrm{i}\int_{0}^{L_{1}}a_{1}(x_{1},x_{2})\mathrm{d}x_{1}}=S,$
$\displaystyle P_{2}(x_{1})$
$\displaystyle=g_{2}(x_{1})\,\mathcal{P}\mathrm{e}^{\mathrm{i}\int_{0}^{L_{2}}a_{2}(x_{1},x_{2})\mathrm{d}x_{2}}=C.$
(20)

Since the classical field is gauge equivalent to $a=0$, the Polyakov loops are
the same as the transition functions. The flatness condition $(F_{12}=0)$
requires $P_{1}P_{3}=P_{3}P_{1}$ and $P_{2}P_{3}=P_{3}P_{2}$, and for a flat
gauge field, we must have

$P_{3}=\mathrm{e}^{2\pi\mathrm{i}m/N}\bm{1},\qquad m=0,1,\ldots,N-1.$ (21)

For this configuration there is no classical energy, but the double trace
deformation (or loop correction) leads to an energy that scales with the size
of $T^{2}$. This is given by

$\displaystyle\Delta E_{\rm
flat}\propto\frac{L_{1}L_{2}}{L_{3}^{2}}\frac{1}{L_{3}}N^{2},$ (22)

In other words, in terms of the energy cost of the two configurations on
$T^{3}=(T^{2})_{12}\times(S^{1})_{3}$, we obtain a classical energy cost for
the non-flat gauge field and a one-loop energy cost for flat gauge field:

$\displaystyle\Delta E_{\rm non-flat}^{\rm
cl.}\propto\frac{N}{\lambda}\left(\frac{L_{3}^{2}}{L_{1}L_{2}}\right)\frac{1}{L_{3}},\qquad\Delta
E_{\rm flat}^{\rm 1-loop}\propto
N^{2}\left(\frac{L_{1}L_{2}}{L_{3}^{2}}\right)\frac{1}{L_{3}},$ (23)

where $\lambda\equiv g^{2}N$ is ’t Hooft coupling. This implies that on
large-$T^{2}\times S^{1}$, it is energetically preferable to turn on a ’t
Hooft flux $n_{12}=1$ by turning on a chromo-magnetic field. Even though a
background chromo-magnetic field costs classical energy, and the cost of the
flat background arises from the 1-loop effect, the background with $F_{12}\neq
0$ is favored for large $L_{1,2}$ energetically.

This implies that on $T^{2}_{\rm large}\times S^{1}_{\rm
small}\times{\mathbb{R}}$, the theory is in an abelianized regime due to non-
trivial gauge holonomy (16). Therefore, the dynamics at distances larger than
inverse $W$-boson mass $(m_{W})^{-1}=\frac{L_{3}N}{2\pi}$ is described as an
abelianized gauge theory

$\displaystyle SU(N)\rightarrow U(1)^{N-1}\qquad P_{3}\;{\rm
as\;adjoint\;Higgs\;field.}$ (24)

Despite the fact that $\mathrm{tr}(P_{3}^{k})=0$ for $k\neq 0$ (mod $N$), the
zero form part of the center-symmetry ${\mathbb{Z}}_{N}^{[0]}$ is _broken_ at
the perturbative level on (8) in an interesting way. The order parameter that
realizes this breaking is $\mathrm{tr}(P_{3}^{k}F_{12})$, the Polyakov loop
with the insertion of $F_{12}$. This operator acquires a non-vanishing
expectation value. In particular,

$\displaystyle\left(\frac{L_{1}L_{2}}{2\pi}\right)\;\mathrm{tr}\left(P_{3}F_{12}^{(i)}\right)=\omega^{i},\quad
i=1,\dots,N.$ (25)

We denote these $N$ classical vacua by $|{\bm{\nu}}_{i}\rangle$, associated
with the weights of the defining representation following (18), and refer to
them as _magnetic flux vacua_

$\displaystyle{\rm
magnetic\;flux\;vacua}:\quad\\{\;|{\bm{\nu}}_{i}\rangle,i=1,\dots,N\\}$ (26)

The theory admits monopole-instantons (which should be viewed as fractional
instantons) interpolating between the perturbative vacua

$\displaystyle|{\bm{\nu}}_{i+1}\rangle\rightarrow|{\bm{\nu}}_{i}\rangle,\qquad\,.$
(27)

The change in the chromo-magnetic flux through tunneling process is
characterized by the magnetic flux of a monopole-instanton.

$\displaystyle Q_{\rm
mag}={\bm{\nu}}_{i}-{\bm{\nu}}_{i+1}\equiv\alpha_{i},\qquad i=1,\ldots,N.$
(28)

The non-zero transition amplitudes

$\displaystyle\langle{\bm{\nu}}_{i}|\mathrm{e}^{-T\widehat{H}_{\mathrm{YM}}}|{\bm{\nu}}_{i+1}\rangle=K\mathrm{e}^{-S_{I}/N+\mathrm{i}\theta/N}$
(29)

between the perturbative vacua leads to the restoration of
$({\mathbb{Z}}_{N}^{[0]})_{3}$ center-symmetry non-perturbatively.

On $\mathbb{R}\times(S^{1})_{\rm small}\times(T^{2})_{\rm large}$ with
$L_{S^{1}}\ll\Lambda^{-1}$, if $L_{T^{2}}$ is asymptotically large, the local
dynamics must be essentially the same as the theory on $\mathbb{R}^{3}\times
S^{1}$. The effect of the ’t Hooft flux insertion via the abelian
chromomagnetic flux (18) has a relatively small effect on local dynamics. The
monopoles and bions due to $SU(N)\rightarrow U(1)^{N-1}$ are present in the
system and their proliferation leads to non-perturbative effects. If $T^{2}$
is very large, then, the proliferation of the monopole-instantons is
mathematically equivalent to the grand canonical ensemble of a magnetic
Coulomb gas in 3d. The magnetic Coulomb gas is most easily described by using
abelian duality, mapping $N-1$ photons into $N-1$ scalars
$*\mathrm{d}{\bm{\sigma}}=L_{3}\frac{2\pi}{\mathrm{i}g^{2}}{\bm{F}}$. The
monopole operator is easier to express in the dual language than the original
description, and they are given by:

$\displaystyle{\cal
M}_{i}(x)=K\mathrm{e}^{-S_{m}}\mathrm{e}^{\mathrm{i}\bm{\alpha}_{i}\cdot\bm{\sigma}(x)}\mathrm{e}^{\mathrm{i}\theta/N}\quad(i=1,\ldots,N),\quad
S_{m}=\frac{S_{I}}{N}\,.$ (30)

Now, we can write the effective field theory action associated with the
magnetic Coulomb gas (See Unsal:2008ch  for details): 666 We use the following
basis frappat:hal-00376660 . Let ${\bm{e}}_{i},\;i=1,\ldots,N$ be an
$N$-component orthonormal basis in ${\mathbb{R}}^{N}$. Define weights of
defining representation
${\bm{\nu}}_{i}={\bm{e}}_{i}-\frac{1}{N}\sum_{j=1}^{N}{\bm{e}}_{j},\;i=1,\ldots,N$.
Then, ${\bm{\nu}}_{i}$ satisfy
${\bm{\nu}}_{i}\cdot{\bm{\nu}}_{j}=\delta_{ij}-1/N$. We define $N-1$
fundamental weights as ${\bm{\mu}}_{j}=\sum_{i=1}^{j}{\bm{\nu}}_{i},j=1,\ldots
N-1$. These ${\bm{\mu}}_{j}$ make a nice basis for dual photons,
${\bm{\sigma}}=\sum_{i=1}^{N-1}\sigma_{i}\bm{\mu}_{i}$, because each
$\sigma_{i}\sim\sigma_{i}+2\pi$ is $2\pi$ periodic. In this basis,
$K_{ij}={\bm{\mu}}_{i}\cdot{\bm{\mu}}_{j}=\min(i,j)-\frac{ij}{N}$ (called
quadratic symmetric form), and in particular,
${\bm{\mu}}_{i}\cdot{\bm{\mu}}_{N-1}=\frac{i}{N}$. $K$ is the inverse of the
Cartan matrix
$A_{ij}={\bm{\alpha}}_{i}\cdot{\bm{\alpha}}_{j}=2\delta_{ij}-\delta_{i,j\pm
1}$, $KA=\bm{1}_{N-1}$.

$\displaystyle S_{\rm 3d}=\int_{{\mathbb{R}}^{3}}{g^{2}\over
8\pi^{2}L_{3}}|\mathrm{d}\bm{\sigma}|^{2}-2\zeta_{m}\sum_{i=1}^{N}\cos\left(\bm{\alpha}_{i}\cdot\bm{\sigma}+{\theta\over
N}\right).$ (31)

This description is valid at energies lower than $W$-boson mass,
$m_{W}=\frac{2\pi}{L_{3}N}$. The inverse of $m_{W}$ also indicates the length
scale associated with monopole size, $r_{\rm m}=\frac{L_{3}N}{2\pi}$. It is
also useful to recall that the dual photon field $\bm{\sigma}$ has periodicity
determined by weight lattice,
$\bm{\sigma}\sim\bm{\sigma}+2\pi\bm{\mu}_{i},\;\bm{\mu}_{i}\in\Gamma_{w}$.

$\displaystyle\bm{\sigma}\in\frac{{\mathbb{R}}^{N-1}}{2\pi\Gamma_{w}}\,.$ (32)

This system generates a non-perturbative mass gap for the dual photon given by
(setting $\theta=0$)

$\displaystyle m_{p}$ $\displaystyle=\Lambda(\Lambda
NL_{3})^{5/6}\sin\frac{\pi p}{N},\qquad p=1,\ldots,N-1,$ (33)
$\displaystyle\ell_{D}$
$\displaystyle=\frac{1}{m_{p=1}}=\frac{N\ell_{YM}}{(\Lambda L_{3}N)^{5/6}}$
(34)

where $\Lambda=\ell_{\rm YM}^{-1}$ is the strong scale of 4d Yang-Mills
theory, and $\ell_{D}$ is the magnetic Debye length. The small parameter
$\epsilon\equiv(\Lambda NL_{3})\ll 1$ controls the semi-classical expansion on
${\mathbb{R}}^{3}\times S^{1}$.

We would like to know the effect of turning on a ’t Hooft flux on this
effective theory. To this end, recall that Yang-Mills theory on
${\mathbb{R}}^{3}\times S^{1}$ has a 1-form and 0-form symmetry that descends
from 4d 1-form symmetry.

$\displaystyle({\mathbb{Z}}_{N}^{[1]})_{4d}\rightarrow({\mathbb{Z}}_{N}^{[1]})_{3d}\times{\mathbb{Z}}_{N}^{[0]}$
(35)

In our first set-up, we turn on a two-form background field (’t Hooft flux)
$B=\frac{2\pi n_{12}}{NL_{1}L_{2}}\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}$,
associated with 1-form symmetry through $T^{2}$. This modifies the 3d action
with an extra topological coupling Tanizaki:2019rbk

$\displaystyle S_{\rm 3d}=\int_{{\mathbb{R}}^{3}}{g^{2}\over
8\pi^{2}L_{3}}|\mathrm{d}\bm{\sigma}|^{2}-\int\frac{\mathrm{i}N}{2\pi}\bm{\mu}_{N-1}\cdot
d{\bm{\sigma}}B-2\zeta_{m}\sum_{i=1}^{N}\cos\left(\bm{\alpha}_{i}\cdot\bm{\sigma}+{\theta\over
N}\right).$ (36)

where $\bm{\mu}_{N-1}=\frac{1}{N}\left(1,1,\ldots,1-N\right)$ is a fundamental
weight. The topological coupling plays a crucial role in our quantum
mechanical set-up.

###  2.2 Separation of scales and transition from 3d to 1d EFT

It is useful to consider the physics of the effective field theory (EFT) on
${\mathbb{R}}\times T^{2}_{\rm large}\times S^{1}_{\rm small}$ in two
different regimes. First of all, because of the compactification, the dual
photon decompose to Kaluza-Klein tower, with characteristic separations
$\frac{2\pi}{L_{T^{2}}}$ between the modes. We would be in QFT regime if

$\displaystyle\frac{2\pi}{L_{T^{2}}}\ll m_{\rm gap}\Rightarrow
L_{T^{2}}\gg\ell_{D}\qquad{\rm 3d\;EFT},$ (37)

so that the dynamics of a dense set of modes generate the non-perturbative 3d
dynamics. In the opposite regime, for

$\displaystyle\frac{2\pi}{L_{T^{2}}}>m_{\rm gap}\Rightarrow
L_{T^{2}}<\ell_{D}\qquad{\rm 1d\;EFT},$ (38)

we can forget about the non-zero KK-modes and consider the dynamics of the
zero mode of the dual photon field $\bm{\sigma}$.

Note that the scale at which quantum mechanical 1d effective field theory
kicks in is not a fixed scale on ${\mathbb{R}}\times T^{2}\times S^{1}$. The
critical value $L_{T^{2}}^{*}$ where this change occurs gets larger and larger
with decreasing $L_{3}$. We can describe the long-distance dynamics via a
quantum mechanical description at $L_{T^{2}}<\frac{N\ell_{YM}}{(\Lambda
L_{3}N)^{5/6}}$.

3D EFT: As long as $T^{2}$ is sufficiently large so that (37) is satisfied one
can use the EFT (31) in order to determine the non-perturbative dynamics. In
particular, the theta dependence of the vacuum energy density can be extracted
from this description. For this purpose we have to find critical points of the
potential in (31), which are possible sets of mean fields for the
${\bm{\sigma}}$ field. There are $N$ constant critical points for
$\bm{\sigma}$ field in its fundamental domain (32). They can be most easily
found by using $\bm{\sigma}=\sum_{i=1}^{N-1}\sigma_{i}{\bm{\mu}}_{i}$. The
potential can then be written as

$\displaystyle
V(\sigma_{i})=-\zeta_{m}(e^{\mathrm{i}\sigma_{1}}+\ldots+e^{\mathrm{i}\sigma_{N-1}}+e^{-\mathrm{i}(\sigma_{1}+\ldots+\sigma_{N-1})}+\rm{c.c})\,.$
(39)

The extrema are given by

$\displaystyle\frac{\partial
V}{\partial\sigma_{i}}=0\quad\Longrightarrow\quad\sigma_{i}=-(\sigma_{1}+\ldots+\sigma_{N-1})\quad\forall
i\qquad\rm{mod}\;2\pi\,.$ (40)

The $N$ critical points can be expressed as

$\displaystyle\bm{\sigma}_{k}^{*}={2\pi\over
N}k\bm{\rho}\,\in\,\frac{{\mathbb{R}}^{N-1}}{2\pi\Gamma_{w}}\,,$ (41)

where $\bm{\rho}=\sum_{i=1}^{N-1}\bm{\mu}_{i}$ is the Weyl vector. These
critical points are shown in Fig. 2 for $N=3$. Since
$\bm{\alpha}_{i}.\bm{\rho}=1,i=1,\ldots,N$, the vacuum energy densities of
these $N$ branches are given by

$\displaystyle\mathcal{E}_{k}(\theta)=-2NK\mathrm{e}^{-S_{I}/N}\cos\left({\theta+2\pi
k\over N}\right).$ (42)

Which $k$ is the true vacuum changes depending on $\theta$, and the $\theta$
dependence of the vacuum energy is given by $\mathcal{E}_{\rm
ground}(\theta)=\min_{k}\mathcal{E}_{k}(\theta)$.

Figure 2: For $SU(3)$ gauge theory, the region bordered by the weight vectors
$\mu_{1},\mu_{2}$ is the fundamental cell of the weight lattice, and it is
also the fundamental domain of ${\bm{\sigma}}/2\pi$ (32). The red marked
points denote the meta-stable vacua of the $N=3$ theory along the Weyl vector
$\rho$ according to (41). These are the mean-fields (critical points) of the
monopole-EFT and as one dials $\theta$-angle, each one will become a genuine
ground state at some value of $\theta$. We show that these are also the
metastable vacua of the quantum mechanical limit of the theory, providing
adiabatic continuity between the two regimes.

###  2.3 Quantum mechanical limit as a deformed TQFT

In the regime $L_{T^{2}}<\ell_{D}$ we can reduce (36) to quantum mechanics.
For this purpose, we have to integrate out all modes except for the zero
momentum mode in the Kaluza-Klein decomposition of the ${\bm{\sigma}}$
field.777 We would like to thank Yuya Tanizaki and Yui Hayashi for pointing
out a flaw in this section in the first version of this manuscript. In the
present construction with abelianized gauge fields the twisted boundary
conditions do not generate a classical mass for the dual photon, i.e. the
classical moduli of sigma are not lifted by the $B$ field. Instead, they
generate a new topological coupling. In §.4, we will see an alternative
construction in which a classical moduli is lifted and a mass for dual photon
is generated. Using $\int_{T^{2}}B=\frac{2\pi n_{12}}{N}$, we see that (36)
reduces to the following quantum mechanical action

$\displaystyle S_{\rm 1d}$ $\displaystyle=S_{\rm kinetic}+S_{\rm
top.coupling}+S_{\rm monopole}\,,$ (43) $\displaystyle S_{\rm 1d}$
$\displaystyle=(L_{1}L_{2})\int_{{\mathbb{R}}}{g^{2}\over
8\pi^{2}L_{3}}\left(\sum_{j=1}^{N-1}K_{ij}\dot{\sigma}_{i}\dot{\sigma}_{j}\right)-\frac{\mathrm{i}n_{12}}{2\pi}\sum_{j=1}^{N-1}\int\frac{2\pi
j}{N}\dot{\sigma}_{j}$ (44)
$\displaystyle-2\zeta_{m}(L_{1}L_{2})\left(\sum_{j=1}^{N-1}\cos\left(\sigma_{j}+{\theta\over
N}\right)+\cos\left(-(\sigma_{1}+\ldots+\sigma_{N-1})+{\theta\over
N}\right)\right)\,,$ (45)

where we used ${\bm{\sigma}}=\sum_{j=1}^{N-1}{\bm{\mu}_{j}}\sigma_{j}$.
$K_{ij}={\bm{\mu}}_{i}\cdot{\bm{\mu}}_{j}=\min(i,j)-\frac{ij}{N}$ is quadratic
symmetric form, (the inverse of the Cartan matrix), and in the topological
term, we used${}^{\ref{conventions}}$
${\bm{\mu}}_{i}\cdot{\bm{\mu}}_{N-1}=\frac{i}{N}$. This is the dimensional
reduction of the dual photon theory with a “topological coupling” induced by
the background 2-form field $B$.888 The same quantum mechanical action also
appears in two other QFTs. One is the compactification of $SU(N)$ level-$k$
WZW model on ${\mathbb{R}}\times S^{1}$ using twisted boundary conditions. The
WZW term gives our topological coupling. The mass deformation of the WZW model
correspond to monopole operators. The other set-up is the fermionization of
this set-up. It is the $N$-flavor Schwinger model with a mass deformation
Misumi:2019dwq , see also Tanizaki:2018xto ; Kikuchi:2017pcp . Note that the
$\theta$ angle appearing in the monopole operator is the theta angle of the 4d
theory. In our set-up, apart from that, the topological coupling to the
background $B$-field generates $N-1$ new theta angles, with values

$\displaystyle\theta_{j}=\frac{2\pi j}{N}\,.$ (46)

These new topological angles play a crucial role in what follows. In
particular, let us first treat $S_{\rm monopole}$ as a small perturbation,
which is justified in this regime. If we set it to zero, then, in the absence
of $\theta_{j}$, the ${\bm{\sigma}}$ fields describe the motion of a particle
on a $T^{N-1}$ torus. In particular, for $N=2$ it describes a free particle on
a circle. The Hamiltonian corresponding to $S_{\rm kinetic}$ in (45) can be
determined by a Legendre transform. We find

$\displaystyle\frac{H}{C}=\sum_{i,j=1}^{N-1}A_{ij}p_{i}p_{j}=\sum_{j=1}^{N-2}(p_{j}-p_{j+1})^{2}+p_{1}^{2}+p_{N-1}^{2}\,,$
(47)

where
$A_{ij}={\bm{\alpha}}_{i}\cdot{\bm{\alpha}}_{j}=2\delta_{ij}-\delta_{i,j\pm
1}$ is the Cartan matrix,
$C=\frac{L_{3}}{2L_{1}L_{2}}\left(\frac{2\pi}{g}\right)^{2}$, and
$p_{i}=\frac{1}{i}\frac{\partial}{\partial\sigma_{i}}$ are the conjugate
momenta. In the absence of $\theta_{j}$, this system has a unique ground
state.

However, we can show that due to the additional theta angles in (46) the non-
interacting theory has an $N$-fold degenerate ground state. $S_{\rm
top.coupling}$ modifies (47) by the replacement $p_{j}\rightarrow
p_{j}-\theta_{j}$:

$\displaystyle\frac{H}{C}=\sum_{i,j=1}^{N-1}A_{ij}\left(p_{i}-n_{12}\frac{i}{N}\right)\left(p_{j}-n_{12}\frac{j}{N}\right)$
(48)

The energy spectrum of the free theory, $E_{\\{m_{n}\\}}$, is given in terms
of $N-1$ quantum numbers, $\\{m_{n}\\}\in{\mathbb{Z}}^{N-1}$. Diagonalizing
the Hamiltonian in the presence of $\theta_{j}$ we find (for $N\geq 3$)

$\displaystyle\frac{E_{\\{m_{n}\\}}}{C}=$
$\displaystyle\sum_{j=1}^{N-2}\left[\left(m_{j}-\frac{n_{12}\,j}{N}\right)-\left(m_{j+1}-\frac{n_{12}(j+1)}{N}\right)\right]^{2}$
(49)
$\displaystyle+\left(m_{1}-\frac{n_{12}}{N}\right)^{2}+\left(m_{N-1}-\frac{n_{12}(N-1)}{N}\right)^{2}\,.$
(50)

For $n_{12}=1$, the ground state is $N$-fold degenerate and the vacua are
given by

$\displaystyle|m_{1},\ldots,m_{N-1}\rangle=\\{|0,\ldots,0\rangle,|0,\ldots,1\rangle,|0,\ldots,1,1\rangle,\ldots|1,\ldots,1\rangle,\\}\,.$
(51)

Note that the ground state energy coincides with (19) as expected.

These vacua are just generalization of the free particle on a circle problem
with theta angle $\theta_{1}=\pi$. In fact, for $N=2$, we have

$\displaystyle E_{m_{1}}$
$\displaystyle={\frac{C}{2}}\left(m_{1}-\frac{1}{2}\right)^{2}\,.$ (52)

The vacua are two fold degenerate and given by $m_{1}=0,1$. In a coordinate
basis, denoting the position of the particle by $\sigma$, the wave functions
are $\langle\sigma|0\rangle={1\over\sqrt{2\pi}}$ and $\langle
0|1\rangle={\mathrm{e}^{\mathrm{i}\sigma}\over\sqrt{2\pi}}$.999 It is
interesting to note that the monopole operators correspond to the wave
function of a particle on a torus, $T^{N-1}$. In the general case the
coordinate space $(\sigma_{1},\ldots,\sigma_{N-1})$ expressions for the states
in the Hilbert space are

$\displaystyle\langle\sigma_{1},\ldots,\sigma_{N-1}|m_{1},\ldots,m_{N-1}\rangle=\mathrm{e}^{\mathrm{i}m_{1}\sigma_{1}}\ldots\mathrm{e}^{\mathrm{i}m_{N-1}\sigma_{N-1}}$
(53)

The coordinate representation also clarifies the role of the monopole
operators. Monopole operators act as

$\displaystyle\langle\tilde{m}_{1},\ldots,\tilde{m}_{N-1}|\mathrm{e}^{\mathrm{i}\sigma_{n}}|m_{1},\ldots,m_{N-1}\rangle=\delta_{\tilde{m}_{1},m_{1}}\ldots\delta_{\tilde{m}_{n},m_{n}+1}\ldots\delta_{\tilde{m}_{N-1},m_{N-1}}\,.$
(54)

We can identify these states with the magnetic flux states described on an
intuitive basis in Unsal:2020yeh . The identification and the role of the
monopole are

$\displaystyle\underbrace{|0,\ldots,0\rangle}_{|{\bm{\nu}}_{N}\rangle}\xrightarrow[]{\mathrm{e}^{\mathrm{i}\sigma_{N-1}}}\underbrace{|0,\ldots,1\rangle}_{|{\bm{\nu}}_{N-1}\rangle}\xrightarrow[]{\mathrm{e}^{\mathrm{i}\sigma_{N-2}}}\underbrace{|0,\ldots,1,1\rangle}_{|{\bm{\nu}}_{N-2}\rangle}\ldots\xrightarrow[]{\mathrm{e}^{\mathrm{i}\sigma_{1}}}\underbrace{|1,\ldots,1\rangle}_{|{\bm{\nu}}_{1}\rangle}\xrightarrow[]{\mathrm{e}^{\mathrm{i}\sigma_{N}}}\underbrace{|0,\ldots,0\rangle}_{|{\bm{\nu}}_{N}\rangle}\,.$
(55)

This shows that the $N-1$ topological couplings in the quantum mechanical
model act as $N-1$ charges (like $\pm{\bm{\nu}}_{1})$ at infinity. In this
way, tunneling happens between $N$ perturbative ground states that share the
same energy.101010 Consider a parallel plate capacitor with charges $Q=\pm 1$.
The energy density inside the capacitor is non-zero and outside, it is zero,
hence they are unequal. Insert a pair of charges at $\pm\infty$, respectively,
$\mp\textstyle{\frac{1}{2}}$. Now, the electric energy density inside the
capacitor and outside are the same. It is well-known in the context of the
Schwinger model that theta angle can have a physical interpretation as charge
at infinity Coleman:1976uz . The topological couplings $\tilde{\theta}_{j}$ or
equivalently, the background magnetic flux ${\bm{\nu}}_{i}$ play the same role
here.

1D EFT, Hamiltonian in terms of magnetic (or electric) flux basis: In the
quantum mechanical description the easiest way to proceed to determine the
spectrum is to use the tight-binding approximation. The representation (55)
makes it clear that we can represent the monopole operators on the space of
perturbative vacua as

$\displaystyle\mathrm{e}^{\mathrm{i}\sigma_{i}}+{\rm
h.c.}\;:\;|{\bm{\nu}}_{i}\rangle\langle{\bm{\nu}}_{i+1}|+{\rm h.c.}$ (56)

The problem reduces (within Born-Oppenheimer approximation) to a particle on a
circle with $N$ degenerate harmonic minima. The tight-binding Hamiltonian is

$\displaystyle
H=\sum_{i=1}^{N}E_{0}|{\bm{\nu}}_{i}\rangle\langle{\bm{\nu}}_{i}|-\sum_{i=1}^{N}K\mathrm{e}^{-S_{m}-\mathrm{i}\theta/N}\
|{\bm{\nu}}_{i}\rangle\langle{\bm{\nu}}_{i+1}|+{\rm h.c.}\,,$ (57)

where the second term describes nearest neighbors hopping between adjacent
flux vacua (55). The first term is the the zero point energy $E_{0}$, which is
the $E_{\rm non-flat}^{\rm cl.}$ in (23). We can easily diagonalize the
Hamiltonian (57) via a discrete Fourier transform:

$\displaystyle|\widetilde{k}\rangle=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\omega^{ki}|{\bm{\nu}}_{i}\rangle$
(58)

and obtain

$\displaystyle
H=\sum_{k=0}^{N-1}{\varepsilon}_{k}(\theta)|\widetilde{k}\rangle\langle\widetilde{k}|$
(59)

where ${\varepsilon}_{k}(\theta)$ is given in (42). We can interpret the
states $|\widetilde{k}\rangle$ as the electric flux states, and describe the
low energy limit of quantum mechanics in terms of either magnetic or electric
basis:

$\displaystyle{\rm
Magnetic\;flux\;states\;}:\;\quad|{\bm{\nu}}_{i}\rangle\quad\Longleftrightarrow\quad|\widetilde{k}\rangle\,:\quad{\rm
Electric\;flux\;states\;}$ (60)

The string tensions for charge $q$ in the the regime $-\pi<\theta<\pi$ are
given by $T_{q}=\varepsilon_{q}(\theta)-\varepsilon_{q}(0)$. The Born-
Oppenheimer appropximation is justified because perturbative gaps are
parametrically larger than the non-perturbative ones.

We observe that the vacuum structure of the theory on $L_{T^{2}}\gg\ell_{D}$
is continuously connected to the one on $L_{T^{2}}\ll\ell_{D}$, and monopole
continues to dictate non=perturbative dynamics even in the quantum mechanical
regime.

###  2.4 Monopole interaction in the QM limit

It is interesting to note that the monopole operator is unchanged in the QM
limit. The interaction between monopoles on ${\mathbb{R}}\times T^{2}_{\rm
large}\times S^{1}_{\rm small}$ or in the ${\mathbb{R}}^{3}\times S^{1}$ limit
is the $1/r$ Coulomb interaction in 3d. Because the photons remain ungapped
classically, the interaction of the monopoles in the quantum mechanical limit
is dictated by the Coulomb interaction in 1d. Inspecting the correlation
functions in the perturbative vacua on ${\mathbb{R}}^{3}\times S^{1}$ and in
$L_{S^{1}}\ll L_{T^{2}}\ll\Lambda^{-1}$ regimes,

$\displaystyle\langle{\cal M}_{i}({\bm{x}}){\cal M}_{j}({\bm{0}})\rangle_{\rm
free,\;3d}$
$\displaystyle=\langle\mathrm{e}^{\mathrm{i}\bm{\alpha}_{i}\cdot\bm{\sigma}({\bm{x}})}\mathrm{e}^{\mathrm{i}\bm{\alpha}_{j}\cdot\bm{\sigma}({\bm{0}})}\rangle$
(61) $\displaystyle\langle{\cal M}_{i}(\tau){\cal M}_{j}(0)\rangle_{\rm
free,\;1d}$
$\displaystyle=\langle\mathrm{e}^{\mathrm{i}\bm{\alpha}_{i}\cdot\bm{\sigma}(\tau)}\mathrm{e}^{\mathrm{i}\bm{\alpha}_{j}\cdot\bm{\sigma}(0)}\rangle\,,$
(62)

we find that the interactions between monopoles in the 3d and 1d EFTs
111111The formula in 1d is valid at distances $\tau\gg L_{T^{2}}/2\pi$, in
EFT. To get a formula which interpolates between 3d interaction and 1d
interaction, we have to incorporate Kaluza-Klein modes. See also the footnote
underneath (102).  are given by

$\displaystyle V({\bm{x}})$
$\displaystyle=A_{ij}\left(\frac{1}{4\pi|{\bm{x}}|}\right)$ (63)
$\displaystyle V(\tau)$
$\displaystyle=\frac{A_{ij}}{(L_{1}L_{2})}\left(-\frac{1}{2}|\tau|\right)$
(64)

where the prefactors

$\displaystyle
A_{ij}=L_{3}\left(\frac{2\pi}{g}\right)^{2}\alpha_{i}\cdot\alpha_{j}\,,$ (65)

arises from the charges of monopoles (with the normalization of kinetic term).

In other words, the monopole-monopole interaction in the quantum mechanical
limit remains long-ranged to all orders in perturbation theory. The system we
have is a multi-component Coulomb gas in $d=1$. The effect of the monopoles
and bions, as it is the case on ${\mathbb{R}}^{3}\times S^{1}$ is to render
the dual photon massive, via magnetic Debye mechanism in 1d. If we insert an
external magnetic charge into the system, its field will decay exponentially
as $\frac{1}{2\kappa}\mathrm{e}^{-\kappa|\tau|}$ where
$\kappa\sim\mathrm{e}^{-S_{m}/2}$ due to Debye screening. In fact, the natural
frequencies of the quantum mechanical system $\omega_{p},=1,\ldots,N-1$ are
exactly the mass spectrum of the theory (34) on ${\mathbb{R}}^{3}\times
S^{1}$, $\omega_{p}=m_{p}$. This shows that the most crucial property of the
magnetic Coulomb gas remains the same in the reduction from $d=3$ to $d=1$.

##  3 Semi-classics on ${\mathbb{R}}\times T^{2}_{\rm small}\times S^{1}_{\rm
large}$ with ’t Hooft flux through $T^{2}$

As it is clear from (23) that the energy costs of having a non-flat field
strength to implement $n_{12}=1$ unit of ’t Hooft flux is proportional to
$\Delta E_{\rm non-flat}\sim(N/\lambda)(L_{3}/L_{T^{2}})^{2}$, while the cost
of implementing it with a flat field strength is proportional to $\Delta
E_{\rm flat}=N^{2}(L_{T^{2}}/L_{3})^{2}$. Therefore, if
$L_{T^{2}}\lesssim\frac{L_{3}}{N^{1/4}\lambda^{1/4}}$, we have to use non-
abelian transition matrices with a flat field strength to probe the vacuum of
the theory. This crossover is very likely related to Nielsen-Olesen
instability as mentioned in Footnote.5, but we postpone this study to future
work.

Assume that $L_{3}\rightarrow\infty$ is very large. Then, our setting is

$\displaystyle M_{4}={\mathbb{R}}^{2}\times
T^{2}=\lim_{L_{3}\rightarrow\infty}\mathbb{R}\times\underbrace{(S^{1})_{3}}_{\rm
large}\times\underbrace{T^{2}}_{\rm short}\,.$ (66)

The short symmetric torus has size $L_{1}=L_{2}\ll\Lambda^{-1}$ and the large
$(S^{1})_{3}$ has size $L_{3}\gg L_{1}$. The semi-classical analysis below is
an overview of necessary ingredients from Tanizaki:2022ngt .

###  3.1 Overview of ${\mathbb{R}}^{2}\times T^{2}_{\rm small}$: 2d EFT is a
deformed TQFT

Introduce $n_{12}=1$ unit of ’t Hooft flux along $T^{2}$. Based on the general
analysis of tHooft:1979rtg , we can choose the transition functions to be
$g_{1}=S$ and $g_{2}=C$ shift and clock matrices. By the relation between
Polyakov loop and transition amplitude (20), this implies that the gauge
holonomies (Polyakov loops) in the $1,2$ directions are given by

$\displaystyle P_{1}=S,\qquad P_{2}=C.$ (67)

Therefore, the theory on ${\mathbb{R}}^{2}\times T^{2}$ may be viewed as a 2d
Yang-Mills theory coupled to two adjoint Higgs fields, $P_{1}$ and $P_{2}$
Tanizaki:2022ngt . In fact, from the twisted Eguchi-Kawai reduction, we know
that the twisted boundary conditions can be replaced by periodic boundary
conditions, but with a modified gauge action GonzalezArroyo:1982ub ;
GonzalezArroyo:1982hz . In our context, for small ${\mathbb{R}}^{2}\times
T^{2}$, this translates to the “classical potential”

$\displaystyle\mathrm{tr}|P_{1}P_{2}-\omega
P_{2}P_{1}|^{2},\qquad\omega=e^{i\frac{2\pi}{N}}\,.$ (68)

The perturbative analysis implies that, similar to the charge $N$ abelian
Higgs model in which the $U(1)$ gauge group is reduced to ${\mathbb{Z}}_{N}$
by the Higgs mechanism, the long distance gauge structure is reduced to
${\mathbb{Z}}_{N}$ gauge theory,

$\displaystyle SU(N)\xrightarrow{P_{1},P_{2}\,\text{adjoint
Higgs}}{\mathbb{Z}}_{N}\,.$ (69)

One can think of this result as an adjoint Higgs regime induced by $P_{1}$
which reduces the $SU(N)$ gauge structure to $U(1)^{N-1}$, followed by a Higgs
mechanism triggered by $P_{2}$ which further reduces $U(1)^{N-1}$ down to
${\mathbb{Z}}_{N}$,

$\displaystyle SU(N)\xrightarrow{P_{1}\,\text{adjoint
Higgs}}U(1)^{N-1}\xrightarrow{P_{2}\,\text{adjoint Higgs}}{\mathbb{Z}}_{N}$
(70)

However, since we are using a symmetric torus, the two Higgs mechanisms occur
at the same energy (length) scale. There is no advantage gained here by
viewing the adjoint Higgs mechanism as a two-step process. However, in the
next section, we will consider an asymmetric torus. There, this perspective
bears some fruits.

Because of the adjoint Higgs mechanism (69), the long distance theory may be
described by a 2D gauge theory with gauge group ${\mathbb{Z}}_{N}$. Moreover,
the twisted boundary conditions force all the gluons to acquire tree-level
masses. We quickly recall how this happens since we will use the result in the
next section too. We can express the gauge fields as

$\displaystyle a=\sum a^{(\bm{p})}J_{\bm{p}}.\quad
J_{\bm{p}}=\omega^{-p_{1}p_{2}/2}C^{-p_{1}}S^{p_{2}},\qquad\bm{p}=(p_{1},p_{2})\in(\mathbb{Z}_{N})^{2}\setminus\\{\bf
0\\}\,,$ (71)

where $J_{\bm{p}}$ is a suitable Lie algebra basis, similar to what is used in
the perturbative analysis of the twisted Eguchi-Kawai model
GonzalezArroyo:1982hz . Denoting the Fourier components of $a^{(\bm{p})}$ by
$a^{(\bm{p,k})}$, where $\bm{k}\in(\mathbb{Z})^{2}$, it is easy to show that
all modes acquire non-zero masses given by

$M_{\bm{p},\bm{k}}^{2}=\left({2\pi\over
NL_{T^{2}}}\right)^{2}\Bigl{(}(Nk_{1}+p_{1})^{2}+(Nk_{2}+p_{2})^{2}\Bigr{)}.$
(72)

Therefore, gluons do not have any zero modes, and the lowest gluon mass is
$2\pi/NL_{T^{2}}$ GonzalezArroyo:1982hz . The spectrum is sketched in Fig. 3
(left). What does this imply for the symmetry realizations?

Recall that in the ${\mathbb{R}}^{2}\times T^{2}$ compactification, the 1-form
symmetry of the 4d theory decompose into 1-form and 0-form symmetries in 2d as

$\displaystyle({\mathbb{Z}}_{N}^{[1]})_{4d}\xrightarrow[]{{\rm
on}\;{{\mathbb{R}}^{2}\times
T^{2}}}({\mathbb{Z}}_{N}^{[1]})_{2d}\times{\mathbb{Z}}_{N}^{[0]}\times({\mathbb{Z}}_{N}^{[0]})\,.$
(73)

Although the 0-form symmetries remain unbroken to all orders in perturbation
theory, the 1-form part of the center symmetry is spontaneously broken, and
the effective 2d theory is perturbatively a topological ${\mathbb{Z}}_{N}$
gauge theory. The main question is, similar to the Polyakov model where in
perturbation theory one lacks linear confinement, whether this infrared limit
is stable or unstable.

Indeed, the proliferation of center-vortices leads to linear confinement. The
center-vortices in 2d proliferate because they are fractional instantons with
topological charge $1/N$ and action $S_{\rm I}/N$ Gonzalez-Arroyo:1998hjb ;
Montero:1999by ; Montero:2000pb , hence their density in the vacuum of the
theory on ${\mathbb{R}}^{2}$ is proportional to $e^{-S_{I}/N}$. Furthermore,
center vortices have a non-trivial mutual statistics with the Wilson loops.
Let ${\bm{x}}\in{\mathbb{R}}^{2}$ be the location of the center of the center-
vortex, and $C$ be the boundary of a disk $D$. Then, Wilson loop
$W_{\cal{R}}(C)$ in the $SU(N)$ representation $\cal{R}$ obeys

$W_{\cal R}(C)=\begin{cases}\omega^{k}&\text{if}\quad{\bm{x}}\in D,\\\
1&\text{if}\quad{\bm{x}}\notin D,\end{cases}$ (74)

where $k$ is the $N$-ality of ${\cal R}$. The center vortex plays two
different roles. The first is as a symmetry generator. In particular, it is
the generator of the 1-form center symmetry Gaiotto:2014kfa . The second is a
dynamical role, their proliferation in 2d set-up is capable of disordering the
Wilson loops, as a result of the phase factors entering the relation (74).

The counterpart of the EFT in 2d for the center-vortex theory is a topological
BF-theory deformed by local topological operators Nguyen:2024ikq ;
Cherman:2021nox .

$\displaystyle S_{\rm 2d}=$ $\displaystyle S_{*}+\Delta
S=\frac{iN}{2\pi}\int_{M}\varphi\,da-2\zeta_{\rm
v}\int_{M}d^{2}x\,\cos(\varphi+\theta/N)\,.$ (75)

Here, $\varphi$ is a compact scalar field ($\varphi\sim\varphi+2\pi$) and $a$
is a $U(1)$ gauge field. The deformation $\Delta S$ describes the
proliferation of center-vortices. The parameter $\zeta_{\rm v}\sim e^{-S_{\rm
I}/N+i\theta/N}$ is the complex fugacity of the center-vortex gas.

The partition function localizes to the $N$ critical points, and is given by

$\displaystyle Z(\theta)=\int D\varphi
Da\;e^{-S}=\sum_{k=0}^{N-1}e^{-\varepsilon_{k}{\cal
A}_{M_{2}}},\qquad\varepsilon_{k}=-2\zeta_{\rm v}\cos\left(\frac{2\pi
k+\theta}{N}\right)\,.$ (76)

The non-degeneracy of the $N$-energy levels implies confinement. To see this,
we need to evaluate the expectation value of the Wilson loop. One finds

$\langle
W^{q}(C)\rangle\sim\exp\\{-(\varepsilon_{k_{0}+q}-\varepsilon_{k_{0}}){\cal
A}\\}.$ (77)

where $k_{0}$ is the ground state for a given range of the $\theta$ angle, and
$\mathcal{A}$ is the area enclosed by $C$. For $\theta\in(-\pi,\pi]$, we find
the string tension associated with the charge $q$ Wilson loop is
$T_{q}=\varepsilon_{q}-\varepsilon_{0}$.

This construction provides a semi-classical realization Tanizaki:2022ngt  of
the center vortex mechanism DelDebbio:1996lih ; DelDebbio:1998luz ;
Engelhardt:1998wu ; Alexandrou:1999iy ; Alexandrou:1999vx ; deForcrand:1999our
; deForcrand:2000pg ; Sale:2022qfn  in a 2d EFT. However, we also note that in
$d\geq 3$, the proliferation of center-vortices is a necessary, but not a
sufficient condition for confinement as shown in Nguyen:2024ikq , while
proliferation of monopoles is sufficient.

###  3.2 ${\mathbb{R}}\times T^{2}_{\rm small}\times S^{1}$

So far, we considered ${\mathbb{R}}^{2}\times T^{2}_{\rm small}$. Now, we
compactify an extra circle, and consider the compactification of the theory on
${\mathbb{R}}\times T^{2}_{\rm small}\times(S^{1})_{3}$. We assume

$\displaystyle L_{T^{2}}=L_{1}=L_{2}\ll L_{3}$ (78)

With this assumption, we can work with flat gauge fields to explore low energy
properties. Let us first describe the classical vacua. As before, we denote
the holonomy of gauge field along long $(S^{1})_{3}$ as $P_{3}$. The classical
vacua are described by configurations of $P_{3}$ that commutes with $P_{1}=S$
and $P_{2}=C$. Since $C$ and $S$ can be used to form all generators of the Lie
algebra of $\mathfrak{su}(N)$, $P_{3}$ must be proportional to identity matrix
$\mathrm{e}^{\mathrm{i}\alpha}\bm{1}$. Since $\det P_{3}=1$, possible values
of $P_{3}=\mathrm{e}^{2\pi\mathrm{i}m/N}\bm{1},\;m=0,1,\ldots,N-1$.

$\displaystyle[P_{3},C]=[P_{3},S]=0,\qquad\det
P_{3}=1\qquad\Longrightarrow\qquad P_{3}=\mathrm{e}^{2\pi\mathrm{i}m/N}\bm{1}$
(79)

Let us denote these classical vacua by $|m\rangle$, obeying

$\displaystyle P_{3}|m\rangle=\mathrm{e}^{2\pi\mathrm{i}m/N}|m\rangle$ (80)

Therefore, perturbatively, $({\mathbb{Z}}_{N}^{[0]})_{3}$ center symmetry is
spontaneously broken.

If one ignores tunneling, one obtains a quantum mechanical system with $N$
vacua. The Hamiltonian at this order is just a constant (it can be set to
$H=0$.) Tunneling events lift the degeneracy of the vacua, similar to (57)

$\displaystyle H=\sum_{m=1}^{N}\tilde{E}_{0}|m\rangle\langle
m|-\sum_{m=1}^{N}K\mathrm{e}^{-S_{\rm v}-\mathrm{i}\theta/N}|m\rangle\langle
m+1|+{\rm h.c.}\,.$ (81)

where the zero point energy $\tilde{E}_{0}$ is the $E_{\rm flat}^{\rm 1-loop}$
in (23). The effective action in this regime is the compactification of the
deformed TQFT (75), and is given by

$\displaystyle S_{\rm 1d}=$ $\displaystyle\frac{iN}{2\pi}\int
dt\varphi\,\frac{d\alpha}{dt}-2(\zeta_{\rm v}L_{3}N)\cos(\varphi+\theta/N)\,,$
(82)

Note, however, that the states entering (81) and (57) look quite different. In
the next subsection, we discuss the two regimes leading to QM more carefully.

###  3.3 Two hierarchies leading to QM, dramatic reordering (same EFT)

In the previous sections we considered two regimes in which the theory on
small ${\mathbb{R}}\times T^{3}$ with $n_{12}=1$ reduces to quantum mechanics.
We studied ${\mathbb{R}}\times T^{2}\times S^{1}$, with both $T^{2}$ and
$S^{1}$ small (smaller than the Yang-Mills length scale $\Lambda^{-1}$), but
with opposite hierarchies of their sizes, $L_{T^{2}}\ll L_{S^{1}}$ vs.
$L_{S^{1}}\ll L_{T^{2}}$. Below we list the physical properties we obtained in
these limits, including the Polyakov loop $P_{3}$, the field strength
$F_{12}$, the perturbative vacua, tunnelings events, tunneling amplitudes, and
the electric flux states and energies. The mapping between these two semi-
classical domains is also examined in Poppitz:2022rxv . The matching of mixed
anomalies with the semi-classical analysis is examined in Tanizaki:2022ngt ;
Cox:2021vsa . With this analysis, we mainly point out that both QM regimes are
described by a ${\mathbb{Z}}_{N}$ topological theory deformed by fractional
instanton operator. The summary of these two regimes is shown in the following
tables.

Regime | Mechanism | Holonomy | Field strength | Pert. vacua | States
---|---|---|---|---|---
$L_{T^{2}}\ll L_{S^{1}}$ | vortices | $\mathrm{tr}P_{3}\propto e^{\mathrm{i}\frac{2\pi}{N}m}$ | $F_{12}=0$ | $\mathrm{tr}(P_{3})\propto e^{\mathrm{i}\frac{2\pi}{N}m}$ | $|m\rangle$
$L_{S^{1}}\ll L_{T^{2}}$ | monopoles | $\mathrm{tr}P_{3}=0$ | $F_{12}=\frac{2\pi H^{(i)}}{gL_{1}L_{2}}$ | $\mathrm{tr}(P_{3}F_{12})\propto e^{\mathrm{i}\frac{2\pi}{N}i}$ | $|{\bm{\nu}}_{i}\rangle$
Tunnelings | Tunneling amp. | vevs | E-flux states | Energies
---|---|---|---|---
$|m\rangle\rightarrow|m+1\rangle$ | ${\rm Exp}[-8\pi^{2}/(g^{2}N)]$ |  $\langle\mathrm{tr}(P_{3})\rangle$=0 | $|\widetilde{q}\rangle\propto\sum_{m}\omega^{qm}|m\rangle$ | $\widetilde{E}_{0}+\varepsilon_{k}(\theta)$
$|{\bm{\nu}}_{i}\rangle\rightarrow|{\bm{\nu}}_{i+1}\rangle$ | ${\rm Exp}[-8\pi^{2}/(g^{2}N)]$ |  $\langle\mathrm{tr}(P_{3}F_{12})\rangle$=0 | $|\widetilde{q}\rangle\propto\sum_{i}\omega^{qi}|{\bm{\nu}}_{i}\rangle$ | $E_{0}+\varepsilon_{k}(\theta)$

We also note that the interaction between vortices is short-ranged while the
ones between monopoles remain long range in the quantum mechanical limit. The
zero point energies $\widetilde{E}_{0}$ is the $E_{\rm flat}^{\rm 1-loop}$ and
$E_{0}$ is the $E_{\rm flat}^{\rm cl.}$ in (23).

In these constructions, the action of center-transformation $U_{\textsf{C}}$
can be used to generate the sum over tunneling amplitudes,
$\mathrm{tr}[e^{-\beta H}(U_{\textsf{ C}})^{n}]$ between
$|m\rangle\rightarrow|m+n\rangle$. One can construct the partition function in
the electric-flux $k$ state by projection, or equivalently, by the Fourier
transform $Z_{k}=\sum_{n=0}^{N-1}\omega^{nk}\mathrm{tr}e^{-\beta
H}(U_{\textsf{C}})^{n}$.

Despite the dramatic change of the Polyakov loop and the field strengths as
$M_{4}$ is dialed from $L_{S^{1}}\ll L_{T^{2}}$ to $L_{T^{2}}\ll L_{S^{1}}$
there is no phase transition, but a reordering of vacua in quantum mechanics.
As explained in Footnote.5, we believe that this dramatic reordering is
related to Nielsen-Olesen instability Nielsen:1978rm , which in turn must be
related to which one of the options in (23) is the lower energy configuration.
For example, for SU(2) gauge theory, the frequencies in mode decomposition of
gluon field with spin $S_{3}=\pm 1$ are Poppitz:2022rxv :

$\displaystyle\omega_{k_{3},n,S_{3}}^{2}(\theta_{12})=\Big{(}\frac{4\pi}{L_{1}L_{2}}(n+{\textstyle{\frac{1}{2}}}+S_{3})+\frac{1}{L_{3}^{2}}(2\pi
k_{3}+(\theta_{12}))^{2}\Big{)},\qquad
n\in{\mathbb{N}}^{0},\;k_{3}\in{\mathbb{Z}}$ (83)

which has a mode that can become tachyonic $(n=0,S_{3}=-1)$. Mainly, in the
$L_{3}\ll L_{T^{2}}$ regime with non-trivial holonomy $\theta_{12}=\pi$,
$\omega_{k_{3},n,S_{3}}^{2}>0$, the would-be tachyonic mode is lifted to
positive values and absent, and $F_{12}\neq 0$ background is stable. In the
$L_{T^{2}}\ll L_{S^{1}}$ regime, with $F_{12}\neq 0$ background,
$\omega_{k_{3},n=0,S_{3}=-1}^{2}(\theta_{12})<0$ is possible, and there is an
instability. Thus, ’t Hooft flux must be implemented without using
chromomagnetic field. This is consistent with two different implementation of
the $n_{12}=1$ ’t Hooft flux in our work.

We indicated this cross-over between the two description in Fig.1 as a red
rearrangement line. The nature of the topological defects in these two regimes
are different. In particular, the one in the $|m\rangle$ basis, are tunneling
events between trivial holonomy vacua $\mathrm{tr}(P_{3})\propto
e^{\mathrm{i}\frac{2\pi}{N}m}$. Only at the core of the center-vortex, the
holonomy becomes non-trivial The ones in the $|{\bm{\nu}}_{i}\rangle$ basis
are tunneling events between nontrivial holonomy vacua, only at the core of
the center-vortex, two eigenvalues of the holonomy becomes degenerate. A
proper understanding of this Nielsen-Olesen cross-over between the two center-
vortex regime is an open question.

Both quantum mechanical limits are described in the deep infrared by a
deformed ${\mathbb{Z}}_{N}$ TQFT in 1d. This is consistent with the idea that
in the diagram Fig. 1, the ${\mathbb{R}}^{2}\times T^{2}$ regime is
adiabatically connected to the ${\mathbb{R}}^{3}\times S^{1}$ regime.

##  4 Two-scale adjoint Higgs mechanism and monopole-vortex continuity: From
3d to 2d

In this section, we give a rederivation of Hayashi-Tanizaki construction on
${\mathbb{R}}^{2}\times S^{1}\times S^{1}$ interpolating directly from 3d to
2d Hayashi:2024yjc . The new result in this section is the 2d EFT (94) in
terms of dual photon variables.

It would be useful if one can recast the adjoint Higgs mechanism to generate a
parametric scale separation between the scale for abelianization
$SU(N)\rightarrow U(1)^{N-1}$ and the scale at which
$U(1)^{N-1}\rightarrow{\mathbb{Z}}_{N}$. Then, one can use the fields of
effective field theory based on the $U(1)^{N-1}$ gauge structure to reach the
${\mathbb{Z}}_{N}$ gauge theory.

In the infinite volume field theory on ${\mathbb{R}}^{d}$, this is rather
standard. We can have two adjoint scalars $(\Phi_{1},\Phi_{2})$ with a
potential favoring a non-commuting minimum, such as the clock and shift
matrices, and arrange a scale separation. This will lead to an abelian theory
at some high energy scale $E_{1}$ and ${\mathbb{Z}}_{N}$ gauge theory at some
lower energy scale $E_{2}$.

$\displaystyle SU(N)\xrightarrow[{\rm scale}\;E_{1}]{\Phi_{1}\,\text{adjoint
Higgs}}\;U(1)^{N-1}\xrightarrow[{\rm scale}\;E_{2}]{\Phi_{2}\,\text{adjoint
Higgs}}\;{\mathbb{Z}}_{N},\qquad E_{2}\ll E_{1}$ (84)

The physics on ${\mathbb{R}}^{d}$ at low-energies $E<E_{2}$ will be described
by ${\mathbb{Z}}_{N}$ gauge theory, intermediate scales $E_{2}<E<E_{1}$ can be
described as a $U(1)^{N-1}$ theory, and high energy $E>E_{1}$ can be described
in terms of the original $SU(N)$ degrees of freedom. All of this takes place
on ${\mathbb{R}}^{d}$.

In our set-ups, very often, a Polyakov loop plays the role of an adjoint Higgs
field. If the Polyakov loop is non-trivial, in the sense that it has non-
degenerate eigenvalues in the weak coupling domain (for example $P=C$, where
$C$ is the clock matrix), then it will lead to abelianization. This happens in
QCD(adj), and in deformed Yang-Mills and QCD. The difference between this and
the adjoint Higgs mechanism that takes place on ${\mathbb{R}}^{d}$ is that on
${\mathbb{R}}^{d-1}\times S^{1}$, with the Polyakov loop acting as a Higgs
field, the long distance theory is defined on ${\mathbb{R}}^{d-1}$, despite
the fact that microscopic theory lives in ${\mathbb{R}}^{d}$.

Therefore, if we can construct a two-stage adjoint Higgs mechanism with non-
commuting Polyakov loops $P_{3}$ and $P_{2}$ then we can achieve the following
hierarchical structure

$\displaystyle SU(N)\;\xrightarrow[{\rm scale}\;L_{3}]{P_{3}\,\text{adjoint
Higgs}}\;\underbrace{U(1)^{N-1}}_{\rm 3d\;EFT}\;\xrightarrow[{\rm
scale}\;L_{2}]{P_{2}\,\text{adjoint
Higgs}}\;\underbrace{{\mathbb{Z}}_{N}}_{\rm 2d\;EFT},\qquad L_{3}\ll L_{2}\,.$
(85)

This asymmetry changes the mass spectrum for the gluonic degrees of freedom
(72) into

$M_{\bm{p},\bm{k}}^{2}=\left({2\pi\over
L_{3}}\right)^{2}\left(k_{3}+\frac{p_{3}}{N}\right)^{2}+\left({2\pi\over
L_{2}}\right)^{2}\left(k_{2}+\frac{p_{2}}{N}\right)^{2}$ (86)

In the limit $L_{2}\rightarrow\infty$ we osberve that $(N-1)$ gluons remain
gapless in perturbation theory, and the $W$-bosons come in $N$-fold degenerate
multiplets, as shown in Fig. 3. At large $L_{2}$ and in the presence of an
$n_{23}=1$ ’t Hooft flux, gapless gluons acquire a tiny mass
$\frac{2\pi}{NL_{2}}$ and the degeneracy among gluons, is lifted as described
by (86).

In particular, the perturbative gaplessness of $N-1$ photons is now lifted
even classically (instead of non-perturbatively by monopoles) into

$M_{{p_{2}},{k_{2}}}^{2}=\left({2\pi\over
L_{2}}\right)^{2}\left(k_{2}+\frac{p_{2}}{N}\right)^{2}$ (87)

due to $n_{23}$ flux. Here, $k_{2}$ is Kaluza-Klein label, and $p_{2}$ is the
(fractional) momentum label.

How does this translate to the dual photon description? To answer this
question, we first have to recall that the 1-form symmetry of the Yang-Mills
theory on ${\mathbb{R}}^{4}$ decomposes in the theory on
${\mathbb{R}}^{3}\times S^{1}$ as

$\displaystyle({\mathbb{Z}}_{N}^{[1]})_{4d}\xrightarrow[]{{\rm
on}\;{{\mathbb{R}}^{3}\times
S^{1}}}({\mathbb{Z}}_{N}^{[1]})_{3d}\times{\mathbb{Z}}_{N}^{[0]}\,.$ (88)

where ${\mathbb{Z}}_{N}^{[0]}$ is the zero form part of the center-symmetry.
Ref. Cherman:2016jtu  pointed out that the gauge invariant definition of the
photon field is
$F_{\mu\nu,j}=\sum_{l=0}^{N-1}\omega^{jl}\mathrm{tr}(P_{3}^{l}F_{\mu\nu})$ and
dual photon is $d\sigma_{j}\propto*F_{\mu\nu,j}$. Since
${\mathbb{Z}}_{N}^{[0]}:P_{3}\rightarrow\omega P_{3}$, this symmetry
cyclically rotates the dual photons as $\sigma_{j}\rightarrow\sigma_{j+1}$,
where we denote the photon in an $N$-component notation
$\bm{\sigma}=(\sigma_{1},\ldots,\sigma_{N})$. The eigenstates of
${\mathbb{Z}}_{N}^{[0]}$ symmetry are same as mass eigenstates of the dual
photon in Unsal:2008ch

$\displaystyle\widetilde{\sigma}_{p}=\frac{1}{\sqrt{N}}\sum_{j}\omega^{jp}\sigma_{j}\,.$
(89)

In this eigenbasis of ${\mathbb{Z}}_{N}^{[0]}$ symmetry, the ’t Hooft twisted
boundary condition translates into

$\displaystyle\widetilde{\sigma}_{p}(x_{0},x_{1},x_{2}+L_{2})=e^{\mathrm{i}\frac{2\pi}{N}p}\widetilde{\sigma}_{p}(x_{0},x_{1},x_{2})\;.$
(90)

Hence, the mode decomposition for the $\widetilde{\sigma}_{p}$ field can be
written as (changing $p$ to $p_{2}$ for convenience)

$\displaystyle\widetilde{\sigma}_{p_{2}}(x_{0},x_{1},x_{2})=\sum_{k_{2}\in{\mathbb{Z}}}\widetilde{\sigma}_{p_{2},k_{2}}(x_{0},x_{1})e^{\mathrm{i}\frac{2\pi}{L_{2}}(k_{2}+\frac{p_{2}}{N})x_{2}}\,,$
(91)

producing the same spectrum as in (87) from a 2d QFT perspective.

Figure 3: (Left) Spectrum of gluons in the presence of a two-stage adjoint
Higgs mechanism based on non-commuting clock and shift matrices. No massless
modes survive. (Right) Spectrum of gluons for a one-stage adjoint Higgs
mechanism with the clock matrix acting as a Higgs field. $N-1$ massless gluons
remain. In the $L_{2}\rightarrow\infty$ limit, the levels in the spectrum on
the left become degenerate and we obtain the spectrum on the right.

Recall that on ${\mathbb{R}}^{3}\times S^{1}$ with center-stable Polyakov loop
$P_{3}$, the long-distance regime is described by the partition function of an
$(N-1)$ component magnetic Coulomb gas whose action is given in (31). As
described in (34), this system has a non-perturbatively induced mass gap
$m_{g}$, or inverse correlation length (magnetic Debye length) $\ell_{D}$. It
is important to recall that $m_{g}$ is non-perturbatively small, or
equivalently, $\ell_{D}$ is very large. Therefore, as we compactify the theory
further down to ${\mathbb{R}}^{2}\times(S^{1})_{2}\times(S^{1})_{3}$, there
are two distinct regimes of the theory because of the interplay between
$L_{2}$ and $\ell_{D}$. These two limits are

$\displaystyle L_{2}\gg\ell_{D}\gg L_{3}\qquad{\rm 3d\;EFT}\,,$ (92)
$\displaystyle\ell_{D}\gg L_{2}\gg L_{3}\qquad{\rm 2d\;EFT}\,.$ (93)

Since the 3d gauge theory is where the semi-classical monopole mechanism of
confinement takes place, and the 2d ${\mathbb{Z}}_{N}$ gauge theory is the
starting point of the 2d center-vortex theory of confinement, this provides an
opportunity to adiabatically connect the two.

First, note that the $(N-1)$ dual photons always become massive. In the regime
$L_{2}\gg\ell_{D}$ the leading order contribution to the mass gap is generated
non-perturbatively and is due to monopole-instantons Unsal:2008ch , and are
given in (34). On the other hand, in the case $L_{2}\lesssim\ell_{D}$, the
mass gap at leading order is a classical one, generated due to the presence of
the $n_{23}=1$ ’t Hooft flux.

Furthermore, due to the twisted boundary condition on the $\bm{\sigma}$ field
in the $L_{2}\ll\ell_{D}$ regime, the $N-1$ continuous moduli are lifted and
$N$ discrete vacua $\bm{\sigma}_{0,k}=\frac{2\pi}{N}{\bm{\rho}}k$ survive as
the classical vacua of the theory. These are indeed the critical points of the
magnetic Coulomb gas on ${\mathbb{R}}^{3}$, and the vacuum structure of the
theory remains unchanged as the origin of the mass gap changes from non-
perturbative monopole induced one to a classical boundary condition induced
one.

For $L_{2}\gg\ell_{D}$ the theory is gapped and confining because of
monopoles. For $L_{2}\ll\ell_{D}$, the theory is classically gapped due to ’t
Hooft flux, and the long distance theory is a ${\mathbb{Z}}_{N}$ gauge theory.
As we described above, this theory allows center-vortices, and effective
action has to be a deformed TQFT in 2d.

The central question is how the 3d effective action (99) based on monopoles
turns into to the deformed TQFT action in 2d? In particular, what is the
precise relation between monopoles and vortices? for large $L_{2}$ vs. smaller
$L_{2}$?

In the regime $L_{2}\ll\ell_{D}$, dual photons have a classical mass,
multiples of $\omega=\frac{2\pi}{NL_{2}}$ due to the ’t Hooft flux. We add the
mass operator in the relevant vacuum. Dimensionally reducing the EFT on
${\mathbb{R}}^{3}\times(S^{1})_{3}$ down to
${\mathbb{R}}^{2}\times(S^{1})_{2}\times(S^{1})_{3}$, we obtain

$\displaystyle S_{\rm 2d}=L_{2}\int_{{\mathbb{R}}^{2}}{g^{2}\over
8\pi^{2}L_{3}}\left(|\mathrm{d}\bm{\sigma}|^{2}+\omega^{2}\min_{k}\left|\bm{\sigma}-\frac{2\pi}{N}{\bm{\rho}}k\right|^{2}\right)-2\zeta_{m}\sum_{a=1}^{N}\cos\left(\bm{\alpha}_{a}\cdot\bm{\sigma}+{\theta\over
N}\right)\,.$ (94)

This is a remarkable formula that incorporates many interesting insights. In
particular, the theory based on variables of abelianized $U(1)^{N-1}$ gauge
theory is equivalent, in the long distance regime, to 2d deformed
${\mathbb{Z}}_{N}$ TQFT. To see this, we note that at large $\omega^{2}$, the
ground state is localized to one of the $N$ gapped critical points
$\bm{\sigma}_{k}^{*}={2\pi\over N}k\bm{\rho},k=0,\ldots,N-1$ along the Weyl
vector (41). Hence, if we use the ansatz ${\bm{\sigma}}={\bm{\rho}}\varphi$,
this leads to a 2d field theory

$\displaystyle S_{\rm 2d}=L_{2}\int_{{\mathbb{R}}}{g^{2}{\bm{\rho}}^{2}\over
8\pi^{2}L_{3}}\left(|\mathrm{d}\varphi|^{2}+\omega^{2}\min_{k}\left|\varphi-\frac{2\pi}{N}k\right|^{2}\right)-2\zeta_{m}N\cos\left(\varphi+{\theta\over
N}\right)\,.$ (95)

The first two terms in this EFT can be recast as a ${\mathbb{Z}}_{N}$ TQFT in
the deep infrared, and the last term is just the center-vortex operator

$\displaystyle S_{\rm 2d}=$
$\displaystyle\frac{iN}{2\pi}\int_{M}\varphi\,da-2\zeta_{\rm
v}\int_{M}d^{2}x\,\cos(\varphi+\theta/N)\,,$ (96)

where we identified $\zeta_{\rm m}L_{2}N\equiv\zeta_{\rm v}$ as the fugacity
of the vortex gas. Note that with these steps, the monopole operator becomes

$\displaystyle
e^{-S_{I}/N}e^{\mathrm{i}\bm{\alpha}_{a}\cdot\bm{\sigma}+\mathrm{i}\frac{\theta}{N}}\rightarrow
L_{2}e^{-S_{I}/N}e^{\mathrm{i}\bm{\alpha}_{a}\cdot\bm{\sigma_{k}}^{*}+\mathrm{i}\frac{\theta}{N}}=L_{2}e^{-S_{I}/N}e^{\mathrm{i}\frac{k}{N}+\mathrm{i}\frac{\theta}{N}}\,,$
(97)

the center-vortex operator in 2d.

At low energies, the mass term forces a localization to the $N$-vacua. This
localization changes the monopole operators to center-vortex induced term
according to (97) in the 2d language, and we obtain the 2d partition function

$\displaystyle Z_{\rm 2d}(\theta)$
$\displaystyle=\sum_{k}e^{V_{{\mathbb{R}}^{2}}2\zeta_{\rm
v}\cos\left(\frac{2\pi k+\theta}{N}\right)}\,,$ (98)

of the semi-classical center-vortex theory in 2d.

We can also consider further compactification of (94) down to quantum
mechanical limit by further compactification of the theory on
${\mathbb{R}}^{2}\times(S^{1})_{2}\times(S^{1})_{3}$ by opening a new circle
on one of the ${\mathbb{R}}^{2}$ directions, say $x_{1}$. This
compactification can be done trivially, by just keeping the zero KK mode of
the $\bm{\sigma}$ field. In this limit, we obtain (recall $n_{23}=1$ now), the
1d effective action:

$\displaystyle S_{\rm 1d}=L_{1}L_{2}\int_{{\mathbb{R}}}{g^{2}\over
8\pi^{2}L_{3}}\left(|\mathrm{d}_{\tau}\bm{\sigma}|^{2}+\omega^{2}\min_{k}\left|\bm{\sigma}-\frac{2\pi}{N}{\bm{\rho}}k\right|^{2}\right)-2\zeta_{m}\sum_{a=1}^{N}\cos\left(\bm{\alpha}_{a}\cdot\bm{\sigma}+{\theta\over
N}\right)\,.$ (99)

It is worthwhile noting that this action looks rather different from the one
we obtained by the quantum mechanical reduction of the theory with $n_{12}=1$,
(45). Yet, both flow to the same deformation of the ${\mathbb{Z}}_{N}$ TQFT.

###  4.1 Monopole interactions and flux fractionalization

How did the magnetic Coulomb gas in 3d turn into a dilute vortex gas in 2d? To
answer this question, we have to understand how monopoles can generate the
effects of center-vortices.

Using (94) we can calculate the interaction between the monopole operators in
this regime. If we were to do a naive dimensional reduction of a 3d Coulomb
gas down to 2d, we would find that the $\pm 1/r$ interactions between magnetic
objects become $\mp\log(r)$ in 2d, see, for example Anber:2011gn ;
Anber:2013doa . However, this is not what we expect in our set-up, which can
be viewed as a _twisted dimensional reduction_. In our construction, we know
that there are center vortices in 2d, and that the interaction between them
must be short ranged, since all the degrees of freedom are gapped. This
property is reminiscent to the distinction between global and gauged vortices,
where in our setup we have to obtain the counter-part of gauged vortices,
similar to gauged abelian Higgs model, see e.g. Manton:2004tk .

The compactification we are considering involves a ’t Hooft flux, which
renders dual photons massive classically. We can inspect the correlation
functions of the monopoles in the perturbative vacuum on
${\mathbb{R}}^{2}\times(S^{1})_{2}\times(S^{1})_{3}$ in the regime (93), in
which the long-distance theory becomes 2d. We find

$\displaystyle\langle{\cal M}_{i}({{x}}){\cal M}_{j}({0})\rangle_{\rm
free,\;2d}$
$\displaystyle=\langle\mathrm{e}^{\mathrm{i}\bm{\alpha}_{i}\cdot\bm{\sigma}({x})}\mathrm{e}^{\mathrm{i}\bm{\alpha}_{j}\cdot\bm{\sigma}({0})}\rangle\,.$
(100)

Assuming that $|{x}|>\frac{NL_{2}}{2\pi}$ where ${x}$ is the separation on
${\mathbb{R}}^{2}$, the interactions are given by

$\displaystyle V({x})$
$\displaystyle=\frac{A_{ij}}{L_{2}}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{e^{\mathrm{i}p.{x}}}{p^{2}+\omega^{2}}=\frac{A_{ij}}{L_{2}}\frac{1}{2\pi}K_{0}(\omega|{x}|)$
(101)
$\displaystyle\approx\frac{A_{ij}}{L_{2}}\frac{1}{4\pi}\left(\frac{NL_{2}}{|{x}|}\right)^{\frac{1}{2}}e^{-\frac{2\pi}{NL_{2}}|{x}|}\quad\quad\left(\omega|{x}|\rightarrow\infty\right)\,,$
(102)

where we have used the asymptotic behavior of the modified Bessel function in
the last step.121212 We can also work without assuming
$|{x}|>(NL_{2})/(2\pi)$. In that case, we have to work with the full 3d Green
function on ${\mathbb{R}}^{2}\times S^{1}$ with twisted boundary condition on
$S^{1}$. This requires modification of the integration as
$\displaystyle\int\frac{d^{2}p}{(2\pi)^{2}}\frac{e^{\mathrm{i}p.x}}{p^{2}+(\frac{2\pi}{L_{2}N})^{2}}\rightarrow\sum_{p_{3}=1}^{N}\sum_{k_{3}\in{\mathbb{Z}}}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{e^{\mathrm{i}p.x}}{p^{2}+(\frac{2\pi}{L_{2}}(k_{3}+\frac{p_{3}}{N}))^{2}}={\sum_{q_{3}\in{\mathbb{Z}}}}^{{}^{\prime}}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{e^{\mathrm{i}p.x}}{p^{2}+(\frac{2\pi
q_{3}}{NL_{2}})^{2}}$ (103) accounting for the Kaluza-Klein modes and the
fractional momentum modes arising from the twisted compactification. In the
last step, we combined the two sums into a single sum, where prime indicates
the absence of zero mode at finite $NL_{2}$. Then, it becomes clear that in
the limit $NL_{2}\rightarrow\infty$ or $|x|\ll NL_{2}$, we obtain 3d Coulomb
interaction $1/(4\pi|x|)$, while at large distances we obtain an exponential
fall-off $\exp[-(2\pi)|x|/(NL_{2})]$. The important point is that the
interaction became short-ranged on ${\mathbb{R}}^{2}$. This implies that the
magnetic flux does not spread on ${\mathbb{R}}^{2}$. It is localized in a
range $\omega^{-1}=\frac{NL_{2}}{2\pi}$. Microscopically, recall that monopole
size is $(m_{W})^{-1}=\frac{NL_{3}}{2\pi}$. To summarize,

$\displaystyle{\rm
monopole\;core\;size\;on\;}{\mathbb{R}}^{2}\times(S^{1})_{2}:$
$\displaystyle\qquad r_{\rm m}=(m_{W})^{-1}=\frac{NL_{3}}{2\pi}$ (104)
$\displaystyle{\rm center-vortex\;size\;on\;}{\mathbb{R}}^{2}:\quad\;$
$\displaystyle\qquad r_{\rm v}=\frac{NL_{2}}{2\pi}$ (105)

Figure 4: (Left) A monopole centered at $x_{2}^{*}$ in
${\mathbb{R}}^{2}\times(S^{1})_{2}$ ($(S^{1})_{3}$ not shown). Due to the ’t
Hooft flux $n_{23}=1$ the monopole flux (which is spherically symmetric at
short distances) collimates into a tube of characteristic size $r_{\rm v}$.
(Right) A Euclidean 2d observable on ${\mathbb{R}}^{2}$ seed this flux as a
center-vortex due to its statistics with Wilson loop.

We can inspect the flux emanating from a monopole centered at $x_{2}^{*}$ in
${\mathbb{R}}^{2}\times(S^{1})_{2}$. We will assume that $N$ is of order
one.131313 The reason we do so here is related to subtleties arising in the
large $N$ limit. Notice that both the microscopic size of a monopole as well
the center-vortex size depends on $N$. This is not an accident. Even at small
$L_{3},L_{2}$, if $N$ is large enough, one can no longer use weak-coupling
semi-classical methods. This is intimately related to large-$N$ volume
independence or (twisted) Eguchi-Kawai reduction Eguchi:1982nm ;
GonzalezArroyo:1982hz ; Kovtun:2007py . The flux is spherically symmetric at
distances much smaller than $L_{2}$. If we consider a sufficiently large
$S^{2}$ surrounding the monopole we find that the flux fractionalizes into two
tubes. ${\bm{\nu}}_{i}$ units of magnetic flux pass through the vicinity of
the North Pole, and $-{\bm{\nu}}_{i+1}$ units of magnetic flux thread the
vicinity of the South pole, see Fig. 4. The total flux is
${\bm{\alpha}}_{i}={\bm{\nu}}_{i}-{\bm{\nu}}_{i+1}$, equal to the monopole
charge.

Remarkably, a Euclidean observable on 2d slice ${\mathbb{R}}^{2}$, will see
either ${\bm{\nu}}_{i+1}$ or ${\bm{\nu}}_{i}$ units of flux. The flux can be
detected using a large Wilson loop in the fundamental representation (larger
than $NL_{2}/2\pi$). Consider a Wilson loop on ${\mathbb{R}}^{2}$,
$W(C)=\frac{1}{N}\,\mathrm{tr}\,e^{\mathrm{i}\oint a}$. In the abelianized
theory, only the photons in the Cartan subalgebra survive and we can write it
as

$\displaystyle W(C)=\frac{1}{N}\,\mathrm{tr}\,e^{\mathrm{i}\oint
a}\xrightarrow{SU(N)\to
U(1)^{N-1}}\frac{1}{N}\sum_{j=1}^{N}e^{\mathrm{i}\oint{\bm{\nu}}_{j}\cdot
a}=\frac{1}{N}\sum_{j=1}^{N}e^{\mathrm{i}\int_{D}{\bm{\nu}}_{j}\cdot f}\;.$
(106)

where $f=\mathrm{d}a$ and $C=\partial D$. Since
${\bm{\nu}}_{i}.{\bm{\nu}}_{j}=\delta_{ij}-\frac{1}{N}$, the flux
$f=2\pi{\bm{\nu}}_{i}$ (for $x_{2}>x_{2}^{*}$) or $f=2\pi{\bm{\nu}}_{i+1}$
(for $x_{2}<x_{2}^{*}$) leads to

$\displaystyle
W(C)=e^{-\mathrm{i}\frac{2\pi}{N}\Theta_{D}(x)}=\left\\{\begin{array}[]{cl}e^{-\mathrm{i}\frac{2\pi}{N}}&\qquad{\rm
vortex}\in D,\cr 1&\qquad{\rm vortex}\notin D.\end{array}\right.$ (109)

where $\Theta_{D}(x)$ is the support function, which is equal to $1$ for $x\in
D$ and $0$ otherwise. In other words, Euclidean 2d observables see the flux
emanating from the monopole as a center-vortex. We refer to this phenomenon as
flux fractionalization. The position of the monopole in the compactified
direction $x_{2}^{*}$ can be interpreted as an internal moduli parameter of
the vortex.

When $L_{2}$ is asymptotically large, then, the a monopole at position
$x\in{\mathbb{R}}^{3}$ with flux (charge) $2\pi\alpha_{j}$ is detected by
abelian Wilson loop with charge ${\bm{\nu}}_{i}$ as

$\displaystyle W_{i}(C)=e^{\mathrm{i}{\bm{\nu}}_{i}\oint
a}=e^{\mathrm{i}{\bm{\nu}}_{i}\int_{D}f}=e^{\mathrm{i}\textstyle{\frac{1}{2}}{\bm{\nu}}_{i}\alpha_{j}\Omega_{D}(x)}=e^{\mathrm{i}\textstyle{\frac{1}{2}}(\delta_{ij}-\delta_{i,j+1})\Omega_{D}(x)}$
(110)

where $\Omega_{D}(x)$ is the solid angle for the oriented surface $D(\partial
D=C)$ subtended at a point $x\in{\mathbb{R}}^{3}$. For an asymptotically large
area $A$, filling a plane ${\mathbb{R}}^{2}$, the solid angle is equal to half
the solid angle of a sphere. If $x$ is below $C$ the solid angle is $2\pi$,
hence

$\displaystyle\lim_{D\rightarrow{\mathbb{R}}^{2}}W_{i}(C)=e^{\mathrm{i}\pi(\delta_{ij}-\delta_{i,j+1})}$
(111)

Then, in the grand-canonical ensemble of monopoles, it is the sum over these
phases that generate the area law of confinement on ${\mathbb{R}}^{3}$. The
main point is that in both the monopole and center-vortex regimes, these
phases are present, and _both_ are sourced by monopoles, and lead to the area
law of confinement.

##  5 Outlook: Metamorphosis

By itself the BPST instanton on ${\mathbb{R}}^{4}$ is a rather simple object.
It is a self-dual field configuration, with action localized in a lump with a
smooth action density. The size of the lump is a moduli parameter.

On $(S^{1})_{3}\times{\mathbb{R}}^{3}$, where the $S^{1}$ is small and has a
center-symmetric holonomy, the lump fractionalizes into $N$ monopole-
instantons with action $\frac{S_{I}}{N}$ and there are long-range interactions
between them. The proliferation of these $N$ monopole-instantons generates a
mass gap (magnetic Debye length $\ell_{D}$) as well as confinement and
fractional theta dependence. These phenomena take place in a semi-classically
calculable regime on $S^{1}\times{\mathbb{R}}^{3}$, which is adiabatically
connected to ${\mathbb{R}}^{4}$. More generally, in QCD-like theories with
fermions, magnetic bions (magnetically charged monopole-antimonopole clusters)
lead to confinement and mass gap.

On $(S^{1})_{3}\times(S^{1})_{2}\times{\mathbb{R}}^{2}$ with a ’t Hooft flux
$n_{23}=1$, and with $L_{3}\ll L_{2}\ll\ell_{D}$, we find that at large
distances on ${\mathbb{R}}^{2}$ monopoles transmute to 2d center-vortices.
These vortices have short-range interaction, and non-trivial mutual statistics
with the Wilson loop due to flux fractionalization. Hence, vortices lead to
confinement and fractional theta dependence in a semi-classically calculable
regime on $T^{2}\times{\mathbb{R}}^{2}$.

Upon further compactification of the center-vortex theory (monopole EFT) to
quantum mechanics on small $T^{3}\times{\mathbb{R}}$, center-vortices
(monopoles) become 1d instantons with short (long) range interaction
interpolating between the $N$-perturbative vacua. This leads to confinement
(as one can deduce from correlators of Polyakov loops) and fractional theta
dependence.

Given that all these scenarios are based on the internal structure of the 4D
instantons when probed by semi-classical fluxes or Polyakov lines, it is
natural to ask whether one can extend these ideas more directly to the four-
dimensional theory.

###### Acknowledgements.

We thank Antonio Gonzalez-Arroyo, Margarita Garcia Perez, Yuya Tanizaki, Yui
Hayashi, David Wandler, Erich Poppitz, Aleksey Cherman, and Mendel Nguyen for
useful discussions. The work is supported by U.S. Department of Energy, Office
of Science, Office of Nuclear Physics under Award Number DE-FG02-03ER41260.

##  Appendix A From charge-$N$ 2d Abelian Higgs model to TQFT

Consider the charge-$N$ Abelian Higgs model with Lagrangian

$\mathcal{L}={1\over 2e^{2}}|f_{12}|^{2}+|(\partial_{\mu}+{\rm
i}Na_{\mu})\Phi|^{2}+\lambda(|\Phi|^{2}-v^{2})^{2}-{{\rm i}\theta\over
2\pi}f_{12},$ (112)

In the classical vacuum, $\Phi=ve^{{\rm i}\xi}$. In the following we set
$v=1$, and ignore the fluctuations of the modulus. The kinetic term for the
scalar field can be expressed as

$\displaystyle|\partial_{\mu}\xi+Na_{\mu}|^{2}\,.$ (113)

We can dualize this Lagrangian by introducing a vector field $B_{\mu}$ and
writing an auxiliary Lagrangian

$\displaystyle\mathcal{L}_{\rm aux}=\frac{1}{4}B_{\mu}^{2}+{\rm
i}\epsilon_{\mu\nu}B_{\mu}(\partial_{\nu}\xi+Na_{\nu})\,.$ (114)

Integrating out $B$ first amounts to setting $B_{\mu}=2{\rm
i}\epsilon_{\mu\nu}(\partial_{\nu}\xi+Na_{\nu})$ and we recover the original
Lagrangian, (113). Instead, if we integrate out $\xi$ first, we obtain
$\epsilon_{\mu\nu}\partial_{\nu}B_{\mu}=0$. Hence, we set
$B_{\mu}=\frac{1}{2\pi}\partial_{\mu}\varphi$ where $\varphi$ is a $2\pi$
periodic field. In terms of the dual variable $\varphi$ the Lagrangian can be
rewritten as

$\mathcal{L}={1\over
2e^{2}}|da|^{2}+\frac{1}{(4\pi)^{2}}|d\varphi|^{2}+\frac{iN}{2\pi}\varphi
da-{{\rm i}\theta\over 2\pi}f_{12}\,.$ (115)

The dual theory has a $U(1)$ gauge field, but there is no field which is
charged under this $U(1)$. Therefore, the gauge field only enters via its
field strength. It is known in the context of the standard abelian Higgs model
that the effect of the gauge field is to generate a mass term for $\varphi$
Coleman:1976uz , and that the IR theory is gapped. However, in the
charge-$N\geq 2$ Abelian Higgs model the IR theory is not completely trivial.
At this stage, we can proceed in two different ways, leading to overlapping
and complementary results.

In the low energy limit $\frac{E}{e}\ll 1$ or equivalently,
$e\rightarrow\infty$, we can drop the gauge kinetic term. The classical
equation of motion for the gauge field $a$ leads to the constraint
$d\varphi=0$, i.e. $\varphi$ becomes non-dynamical. To all orders in
perturbation theory, the effective 2d theory at large distances is the
topological ${\mathbb{Z}}_{N}$ gauge theory (TQFT). The action of the
${\mathbb{Z}}_{N}$ TQFT can be expressed as Banks:2010zn ; Kapustin:2014gua

$S_{*}=\frac{iN}{2\pi}\int_{M}\varphi\,da,$ (116)

The fundamental question at this stage, then, is whether this infrared limit
is stable or unstable. We will see that with the inclusion of non-perturbative
effects (instantons, which are vortices in this set-up) is crucial in this
context.

Before answering this question, we note that in the TQFT limit the theory no
longer contains the mass of the $\varphi$ fluctuations. In order to establish
the relation with the statistical field theory that describes the vacuum of
the model, it is useful to keep the fluctuating massive scalars in the IR
description.

We can keep the mass of the scalar $\varphi$ using the following procedure.
Since there is no charged matter field in the dual description, the appearance
of the gauge field is through the field strength $f=da$. If we view the theory
on ${\mathbb{R}}^{2}$ as the decompactification limit of the theory on some
$M_{2}$, we can replace the integral over $a$ with an integral over $f$ with
the inclusion of a sum over all flux sectors, i.e, $\int Da\rightarrow\int
Df\sum_{\nu\in{\mathbb{Z}}}\delta\left(\nu-\frac{1}{2\pi}\int f\right)$. The
latter integral can further be replaced with $\int
Df\sum_{n\in{\mathbb{Z}}}e^{{\rm i}n\int f}$ by using the Poisson resummation
formula. As a result, we can integrate out $f$ at a finite value of $e^{2}$ to
obtain

$\displaystyle\int Da\;{\rm exp}\left[-{1\over 2e^{2}}\int|f|^{2}+\frac{{\rm
i}}{2\pi}N\int\varphi f+\frac{{\rm i}}{2\pi}\theta\int f\right]$ (117)
$\displaystyle=\int Df\;\sum_{n\in{\mathbb{Z}}}\;{\rm exp}\left[-{1\over
2e^{2}}\int|f|^{2}+\frac{{\rm i}N}{2\pi}\int\left(\varphi+\frac{\theta+2\pi
n}{N}\right)f\right]$ (118) $\displaystyle=C\;{\rm
exp}\left[-\frac{1}{2}\frac{e^{2}N^{2}}{4\pi^{2}}\int\min_{n\in{\mathbb{Z}}}\left(\varphi+\frac{\theta+2\pi
n}{N}\right)^{2}\right]\,,$ (119)

where $C$ is a constant. Therefore, the dual Lagrangian to all orders in
perturbation theory takes the form

$\displaystyle\mathcal{L}=\frac{1}{(4\pi)^{2}}|d\varphi|^{2}+\frac{1}{2}\frac{e^{2}N^{2}}{4\pi^{2}}\min_{n\in{\mathbb{Z}}}\left(\varphi+\frac{\theta+2\pi
n}{N}\right)^{2}\,,$ (120)

i.e. $\varphi$ is free massive scalar. Within the fundamental domain of
$\varphi$, the theory has $N$ minima. Setting $\theta=0$ these minima are
located at

$\displaystyle\varphi_{*}=\frac{2\pi n}{N},\;\;n=0,1,\ldots,N-1\,.$ (121)

The massive theory, (120), in the deep infrared, reduces to TQFT. Indeed, the
minima (121) are consistent with the TQFT action, for which the equation of
motion of $a$ yields $d\varphi=0$, and the possible values of $\varphi$ are
given by the same set of discrete points.

The inclusion of vortices amounts to deforming (116) or (120) as

$\displaystyle S$
$\displaystyle=\frac{iN}{2\pi}\int_{M}\varphi\,da-2\zeta_{\rm
v}\int_{M}\cos{(\varphi+{\theta}/{N}})$ (122) $\displaystyle S$
$\displaystyle=\int_{M}\frac{1}{(4\pi)^{2}}|d\varphi|^{2}+\frac{1}{2}m_{\varphi}^{2}``\varphi^{2}"-2\zeta_{\rm
v}\int_{M}\cos({\varphi+{\theta}/{N})}\,,$ (123)

where the mass term $m_{\varphi}^{2}``\varphi^{2}"$ has to be understood as in
(120) and we shifted $\varphi$ to have the $\theta$ angle in the vortex term.

There is however, a disadvantage to working with the deformed TQFT. This EFT
does not account for the interaction between vortices, and treats them as non-
interacting. This is justified because the interaction deduced from (123) by
inspecting the vortex-vortex correlators is:

$\displaystyle\langle
e^{\mathrm{i}\varphi(x)}e^{\pm\mathrm{i}\varphi(0)}\rangle_{\rm
free}=e^{-V(x)}\quad\Longrightarrow\quad V(x)\sim\pm
K_{0}(m_{\varphi}|x|)\sim\pm e^{-m_{\varphi}|x|}$ (124)

The TQFT limit corresponds to $m_{\varphi}\rightarrow\infty$, and the
interactions does indeed disappear. However, in our story, in order to explain
how the long-range interaction between monopoles transmutes to the short-range
interaction among vortices, it is very important to keep track of the
interactions. More generally, it is useful to keep track of the interactions
between topological configurations when we map the Euclidean vacuum structure
of a theory to statistical field theory.

##  Appendix B An electrostatic analogy: Flux fractionalization

In the case of $SU(N)$ gauge theory Hayashi and Tanizaki pointed out that the
relation between monopoles in a ’t Hooft flux background and center-vortices
can be mapped to an electro-static problem Hayashi:2024yjc . Here, we detail
the simplest realization of this to $SU(2)$ gauge theory in detail. The goal
is to point out that the potential becomes short range, exponentially decaying
away from the chain. This is in agreement with our findings (102) obtained by
evaluating monopole-correlators in $n_{12}=1$ background.

Consider an infinite line array of alternating charges, $\pm$, separated from
each other by the spacing $L_{2}$. We can compute the electrostatic potential
(in cylindrical coordinates) at position $\vec{\rho}\equiv(x_{0},x_{1})$ and
$x_{2}$. Let us position $\pm$ charges at $(x_{2}^{*}+2L_{2}k$,
$(x_{2}^{*}+2L_{2}(k+\textstyle{\frac{1}{2}}))$, respectively, and shift the
coordinates such that $x_{2}^{*}=0$ for convenience.141414In the microscopic
theory, the $+$ charge comes from the BPS monopole and $-$ charge arises from
KK monopoles, both with $Q_{T}=\textstyle{\frac{1}{2}}$ topological charge,
and $e^{\mathrm{i}\frac{\theta}{2}}$ theta angle dependence. It is important
to note that $-$ charge is not an anti-monopole. In the case of a general
$SU(N)$ group twisted boundary condition imply the presence of a chain of
monopoles
$\ldots\alpha_{1}\alpha_{2}\ldots\alpha_{N}\alpha_{1}\alpha_{2}\ldots\alpha_{N}\ldots$
repeating indefinitely, each monopole with $e^{\mathrm{i}\frac{\theta}{N}}$
theta dependence.

Figure 5: (Left) An alternating array of $\pm$ charges. The potential
evaluated at $\rho,x_{2}$ decays exponentially for large $\rho$ as
$e^{-\frac{2\pi}{2L_{2}}\rho}$. (Right) The flux of a charged particle
collimates into a tube of thickness $2L_{2}/2\pi$. The fractionalized flux
gives a contribution $W(C)=e^{\mathrm{i}\pi}$ to a Wilson loop on
${\mathbb{R}}^{2}$, leading to area law of confinement on
${\mathbb{R}}^{2}\times T^{2}$.

This is an interesting problem, because each type of charge array $(\pm)$
individualy produces a $\mp\log\rho$ term at large distances. But of course,
due to the alternating charges, we expect $V(\rho,x_{2})\rightarrow 0$ as
$\rho\rightarrow\infty$. But the question is how does the potential decay
exactly, algebraically or exponentially? In the analysis in the bulk of the
paper, we reduced the 3d Coulomb gas EFT down to a deformed TQFT in 2d. This
implies that the fall-off must be exponential. We would like to see that the
exponential decay of the potential emerges from a sum of the type
$\sum_{k\in{\mathbb{Z}}}\frac{Q_{k}}{|r-r_{k}|}$.

The potential at some $(\rho,x_{2})$ can be written as

$\displaystyle
V(\rho,x_{2})=\frac{1}{4\pi}\sum_{k\in{\mathbb{Z}}}\frac{1}{\sqrt{(x_{2}-2L_{2}k)^{2}+\rho^{2}}}-\frac{1}{4\pi}\sum_{k\in
Z}\frac{1}{\sqrt{(x_{2}-2L_{2}(k+\textstyle{\frac{1}{2}}))^{2}+\rho^{2}}}\,.$
(125)

Clearly, $V(\rho,x_{2})$ is periodic with period $2L_{2}$ in the $x_{2}$
direction, and $V(\rho,x_{2})=V(\rho,-x_{2})$. Therefore, we can Fourier
expand $V(\rho,x_{2})$ in the orthogonal eigenbasis
$\\{\cos(\frac{2\pi}{2L_{2}}mx_{2}),\;m=[0,\infty)\\}$ as

$\displaystyle
V(\rho,x_{2})=\sum_{m=0}^{\infty}V_{m}(\rho)\cos\left(\frac{2\pi}{2L_{2}}mx_{2}\right)\,.$
(126)

The Fourier coefficients $V_{m}(\rho)$ tells us how each mode behaves as a
function of $\rho$, and can be found via inverse transform

$\displaystyle V_{m}(\rho)$
$\displaystyle=\frac{1}{2L_{2}}\int_{-L_{2}}^{+L_{2}}dx_{2}\,V(\rho,x_{2})\cos\left(\frac{2\pi}{2L_{2}}mx_{2}\right),$
(127)
$\displaystyle=\frac{1}{2L_{2}}\left(1-(-1)^{m}\right)\int_{-\infty}^{+\infty}dx_{2}\,\frac{1}{4\pi\sqrt{{(x_{2})^{2}+\rho^{2}}}}\cos\left(\frac{2\pi}{2L_{2}}mx_{2}\right)\,.$
(128)

Remarkably, the Fourier coefficient of the zero mode $m=0$ vanishes, as well
as all even modes, $m\in 2{\mathbb{Z}}^{+}$. We could have anticipated this
easily because the potential at $x_{2}=L_{3}/2+L_{3}{\mathbb{Z}}$ must vanish
at any $\rho$ by symmetry. For $m\in 2{\mathbb{Z}}^{+}+1$, we obtain the
Fourier coefficients

$\displaystyle V_{m}(\rho)$ $\displaystyle=\frac{1}{2\pi
L_{2}}K_{0}\left(\frac{2\pi}{2L_{2}}m\rho\right)\,.$ (129)

This implies that the potential has the form

$\displaystyle V(\rho,x_{2})=\frac{1}{2\pi
L_{2}}\sum_{m=1,3,\ldots}K_{0}\left(\frac{2\pi}{2L_{2}}m\rho\right)\cos\left(\frac{2\pi}{2L_{2}}mx_{2}\right),$
(130)

At short distances, close to the cores of monopoles, this potential behaves as
$\pm 1/r$ as it should. At long distances, all the modes decay exponentially
fast for $\rho>\frac{2L_{2}}{2\pi}$. Even the slowest decaying mode falls off
as $e^{-\frac{2\pi}{2L_{2}}\rho}$. This is the same result as we obtained in
(102) from field theory.

## References

  * (1) M. Unsal, “Abelian duality, confinement, and chiral symmetry breaking in QCD(adj),” Phys. Rev. Lett. 100 (2008) 032005, arXiv:0708.1772 [hep-th].
  * (2) M. Unsal, “Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions,” Phys. Rev. D80 (2009) 065001, arXiv:0709.3269 [hep-th].
  * (3) M. Unsal and L. G. Yaffe, “Center-stabilized Yang-Mills theory: Confinement and large N volume independence,” Phys. Rev. D78 (2008) 065035, arXiv:0803.0344 [hep-th].
  * (4) M. Shifman and M. Unsal, “QCD-like Theories on R(3) x S(1): A Smooth Journey from Small to Large r(S(1)) with Double-Trace Deformations,” Phys. Rev. D78 (2008) 065004, arXiv:0802.1232 [hep-th].
  * (5) N. M. Davies, T. J. Hollowood, and V. V. Khoze, “Monopoles, affine algebras and the gluino condensate,” J. Math. Phys. 44 (2003) 3640–3656, arXiv:hep-th/0006011 [hep-th].
  * (6) E. Poppitz, “Notes on Confinement on $R^{3}\times S^{1}$: From Yang-Mills, Super-Yang-Mills, and QCD (adj) to QCD(F),” Symmetry 14 no. 1, (2022) 180, arXiv:2111.10423 [hep-th].
  * (7) Y. Tanizaki and M. Ünsal, “Center vortex and confinement in Yang–Mills theory and QCD with anomaly-preserving compactifications,” PTEP 2022 no. 4, (2022) 04A108, arXiv:2201.06166 [hep-th].
  * (8) Y. Tanizaki and M. Ünsal, “Semiclassics with ’t Hooft flux background for QCD with 2-index quarks,” JHEP 08 (2022) 038, arXiv:2205.11339 [hep-th].
  * (9) Y. Hayashi, Y. Tanizaki, and H. Watanabe, “Semiclassical analysis of the bifundamental QCD on $\mathbb{R}^{2}\times T^{2}$ with ’t Hooft flux,” JHEP 10 (2023) 146, arXiv:2307.13954 [hep-th].
  * (10) Y. Hayashi and Y. Tanizaki, “Semiclassics for the QCD vacuum structure through $T^{2}$-compactification with the baryon-’t Hooft flux,” arXiv:2402.04320 [hep-th].
  * (11) E. Poppitz, T. Schäfer, and M. Ünsal, “Continuity, Deconfinement, and (Super) Yang-Mills Theory,” JHEP 10 (2012) 115, arXiv:1205.0290 [hep-th].
  * (12) Y. Hayashi, Y. Tanizaki, and H. Watanabe, “Non-supersymmetric duality cascade of QCD(BF) via semiclassics on $\mathbb{R}^{2}\times T^{2}$ with the baryon-’t Hooft flux,” arXiv:2404.16803 [hep-th].
  * (13) E. Poppitz and F. D. Wandler, “Gauge theory geography: charting a path between semiclassical islands,” JHEP 02 (2023) 014, arXiv:2211.10347 [hep-th].
  * (14) A. A. Cox, E. Poppitz, and F. D. Wandler, “The mixed 0-form/1-form anomaly in Hilbert space: pouring the new wine into old bottles,” JHEP 10 (2021) 069, arXiv:2106.11442 [hep-th].
  * (15) M. M. Anber and E. Poppitz, “Multi-fractional instantons in $SU(N)$ Yang-Mills theory on the twisted $\mathbb{T}^{4}$,” JHEP 09 (2023) 095, arXiv:2307.04795 [hep-th].
  * (16) M. Yamazaki, “Relating ’t Hooft Anomalies of 4d Pure Yang-Mills and 2d $\mathbb{CP}^{N-1}$ Model,” JHEP 10 (2018) 172, arXiv:1711.04360 [hep-th].
  * (17) M. Yamazaki and K. Yonekura, “From 4d Yang-Mills to 2d $\mathbb{CP}^{N-1}$ model: IR problem and confinement at weak coupling,” JHEP 07 (2017) 088, arXiv:1704.05852 [hep-th].
  * (18) T. C. Kraan and P. van Baal, “Periodic instantons with nontrivial holonomy,” Nucl. Phys. B533 (1998) 627–659, arXiv:hep-th/9805168 [hep-th].
  * (19) T. C. Kraan and P. van Baal, “Monopole constituents inside SU(n) calorons,” Phys. Lett. B435 (1998) 389–395, arXiv:hep-th/9806034 [hep-th].
  * (20) K.-M. Lee and C.-h. Lu, “SU(2) calorons and magnetic monopoles,” Phys. Rev. D58 (1998) 025011, arXiv:hep-th/9802108 [hep-th].
  * (21) K.-M. Lee and P. Yi, “Monopoles and instantons on partially compactified D-branes,” Phys. Rev. D56 (1997) 3711–3717, arXiv:hep-th/9702107 [hep-th].
  * (22) A. Gonzalez-Arroyo and A. Montero, “Selfdual vortex - like configurations in SU(2) Yang-Mills theory,” Phys. Lett. B 442 (1998) 273–278, arXiv:hep-th/9809037.
  * (23) A. Montero, “Study of SU(3) vortex - like configurations with a new maximal center gauge fixing method,” Phys. Lett. B 467 (1999) 106–111, arXiv:hep-lat/9906010.
  * (24) A. Montero, “Vortex configurations in the large N limit,” Phys. Lett. B 483 (2000) 309–314, arXiv:hep-lat/0004002.
  * (25) A. Gonzalez-Arroyo, “On the fractional instanton liquid picture of the Yang-Mills vacuum and Confinement,” arXiv:2302.12356 [hep-th].
  * (26) A. M. Polyakov, “Compact Gauge Fields and the Infrared Catastrophe,” Phys. Lett. B59 (1975) 82–84.
  * (27) A. Polyakov, Gauge Fields and Strings (Mathematical Reports,).  CRC Press, 1 ed., 9, 1987.
  * (28) T. Banks and N. Seiberg, “Symmetries and Strings in Field Theory and Gravity,” Phys. Rev. D83 (2011) 084019, arXiv:1011.5120 [hep-th].
  * (29) A. Kapustin and N. Seiberg, “Coupling a QFT to a TQFT and Duality,” JHEP 04 (2014) 001, arXiv:1401.0740 [hep-th].
  * (30) M. Nguyen, T. Sulejmanpasic, and M. Ünsal, “Phases of theories with $\mathbb{Z}_{N}$ 1-form symmetry and the roles of center vortices and magnetic monopoles,” arXiv:2401.04800 [hep-th].
  * (31) A. Cherman, T. Jacobson, and M. Neuzil, “Universal Deformations,” SciPost Phys. 12 no. 4, (2022) 116, arXiv:2111.00078 [hep-th].
  * (32) Y. Hayashi and Y. Tanizaki, “Unifying Monopole and Center Vortex as the Semiclassical Confinement Mechanism,” arXiv:2405.12402 [hep-th].
  * (33) G. ’t Hooft, “A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories,” Nucl. Phys. B153 (1979) 141–160.
  * (34) G. ’t Hooft, “Aspects of Quark Confinement,” Phys. Scripta 24 (1981) 841–846.
  * (35) P. van Baal, “Some Results for SU(N) Gauge Fields on the Hypertorus,” Commun. Math. Phys. 85 (1982) 529.
  * (36) L. Del Debbio, M. Faber, J. Greensite, and S. Olejnik, “Center dominance and Z(2) vortices in SU(2) lattice gauge theory,” Phys. Rev. D 55 (1997) 2298–2306, arXiv:hep-lat/9610005.
  * (37) L. Del Debbio, M. Faber, J. Giedt, J. Greensite, and S. Olejnik, “Detection of center vortices in the lattice Yang-Mills vacuum,” Phys. Rev. D 58 (1998) 094501, arXiv:hep-lat/9801027.
  * (38) M. Engelhardt, K. Langfeld, H. Reinhardt, and O. Tennert, “Interaction of confining vortices in SU(2) lattice gauge theory,” Phys. Lett. B 431 (1998) 141–146, arXiv:hep-lat/9801030.
  * (39) C. Alexandrou, M. D’Elia, and P. de Forcrand, “The Relevance of center vortices,” Nucl. Phys. B Proc. Suppl. 83 (2000) 437–439, arXiv:hep-lat/9907028.
  * (40) C. Alexandrou, P. de Forcrand, and M. D’Elia, “The Role of center vortices in QCD,” Nucl. Phys. A 663 (2000) 1031–1034, arXiv:hep-lat/9909005.
  * (41) P. de Forcrand and M. D’Elia, “On the relevance of center vortices to QCD,” Phys. Rev. Lett. 82 (1999) 4582–4585, arXiv:hep-lat/9901020.
  * (42) P. de Forcrand and M. Pepe, “Center vortices and monopoles without lattice Gribov copies,” Nucl. Phys. B 598 (2001) 557–577, arXiv:hep-lat/0008016.
  * (43) N. Sale, B. Lucini, and J. Giansiracusa, “Probing center vortices and deconfinement in SU(2) lattice gauge theory with persistent homology,” Phys. Rev. D 107 no. 3, (2023) 034501, arXiv:2207.13392 [hep-lat].
  * (44) J. Greensite, “The Confinement problem in lattice gauge theory,” Prog. Part. Nucl. Phys. 51 (2003) 1, arXiv:hep-lat/0301023.
  * (45) S. Deldar and S. M. Hosseini Nejad, “Center vortices as composites of monopole fluxes,” AIP Conf. Proc. 1701 no. 1, (2016) 100004, arXiv:1503.04459 [hep-ph].
  * (46) L. E. Oxman, “4D ensembles of percolating center vortices and monopole defects: The emergence of flux tubes with N -ality and gluon confinement,” Phys. Rev. D 98 no. 3, (2018) 036018, arXiv:1805.06354 [hep-th].
  * (47) G. ’t Hooft, “On the phase transition towards permanent quark confinement,” Nucl.Phys.B 138 (1978) 1–25.
  * (48) J. M. Cornwall, “Quark Confinement and Vortices in Massive Gauge Invariant QCD,” Nucl. Phys. B 157 (1979) 392–412.
  * (49) H. B. Nielsen and P. Olesen, “A Quantum Liquid Model for the QCD Vacuum: Gauge and Rotational Invariance of Domained and Quantized Homogeneous Color Fields,” Nucl. Phys. B 160 (1979) 380–396.
  * (50) D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized Global Symmetries,” JHEP 02 (2015) 172, arXiv:1412.5148 [hep-th].
  * (51) M. Ünsal, “Strongly coupled QFT dynamics via TQFT coupling,” JHEP 11 (2021) 134, arXiv:2007.03880 [hep-th].
  * (52) M. García Pérez, A. González-Arroyo, and M. Okawa, “Spatial volume dependence for 2+1 dimensional SU(N) Yang-Mills theory,” JHEP 09 (2013) 003, arXiv:1307.5254 [hep-lat].
  * (53) D. J. Gross, R. D. Pisarski, and L. G. Yaffe, “QCD and Instantons at Finite Temperature,” Rev. Mod. Phys. 53 (1981) 43.
  * (54) P. Kovtun, M. Unsal, and L. G. Yaffe, “Volume independence in large N(c) QCD-like gauge theories,” JHEP 06 (2007) 019, arXiv:hep-th/0702021 [HEP-TH].
  * (55) A. Cherman, M. Shifman, and M. Unsal, “Bose-Fermi cancellations without supersymmetry,” Phys. Rev. D99 no. 10, (2019) 105001, arXiv:1812.04642 [hep-th].
  * (56) M. Unsal and L. G. Yaffe, “Large-N volume independence in conformal and confining gauge theories,” JHEP 08 (2010) 030, arXiv:1006.2101 [hep-th].
  * (57) N. K. Nielsen and P. Olesen, “An Unstable Yang-Mills Field Mode,” Nucl. Phys. B 144 (1978) 376–396.
  * (58) L. Frappat, P. Sorba, and A. Sciarrino, Dictionary on Lie algebras and superalgebras.  Academic Press (London), 2000.  https://hal.science/hal-00376660.
  * (59) Y. Tanizaki and M. Unsal, “Modified instanton sum in QCD and higher-groups,” JHEP 03 (2020) 123, arXiv:1912.01033 [hep-th].
  * (60) T. Misumi, Y. Tanizaki, and M. Ünsal, “Fractional $\theta$ angle, ’t Hooft anomaly, and quantum instantons in charge-$q$ multi-flavor Schwinger model,” JHEP 07 (2019) 018, arXiv:1905.05781 [hep-th].
  * (61) Y. Tanizaki and T. Sulejmanpasic, “Anomaly and global inconsistency matching: $\theta$-angles, $SU(3)/U(1)^{2}$ nonlinear sigma model, $SU(3)$ chains and its generalizations,” Phys. Rev. B98 no. 11, (2018) 115126, arXiv:1805.11423 [cond-mat.str-el].
  * (62) Y. Kikuchi and Y. Tanizaki, “Global inconsistency, ’t Hooft anomaly, and level crossing in quantum mechanics,” Prog. Theor. Exp. Phys. 2017 (2017) 113B05, arXiv:1708.01962 [hep-th].
  * (63) S. R. Coleman, “More About the Massive Schwinger Model,” Annals Phys. 101 (1976) 239.
  * (64) A. Gonzalez-Arroyo and M. Okawa, “A Twisted Model for Large $N$ Lattice Gauge Theory,” Phys. Lett. 120B (1983) 174–178.
  * (65) A. Gonzalez-Arroyo and M. Okawa, “The Twisted Eguchi-Kawai Model: A Reduced Model for Large N Lattice Gauge Theory,” Phys. Rev. D27 (1983) 2397.
  * (66) A. Cherman and E. Poppitz, “Emergent dimensions and branes from large-$N$ confinement,” Phys. Rev. D 94 no. 12, (2016) 125008, arXiv:1606.01902 [hep-th].
  * (67) M. M. Anber, E. Poppitz, and M. Unsal, “2d affine xy-spin model/4d gauge theory duality and deconfinement,” JHEP 04 (2012) 040, arXiv:1112.6389 [hep-th].
  * (68) M. M. Anber, S. Collier, E. Poppitz, S. Strimas-Mackey, and B. Teeple, “Deconfinement in $\mathcal{N}=1$ super Yang-Mills theory on $\mathbb{R}^{3}\times\mathbb{S}^{1}$ via dual-Coulomb gas and ”affine” XY-model,” JHEP 11 (2013) 142, arXiv:1310.3522 [hep-th].
  * (69) N. S. Manton and P. Sutcliffe, Topological solitons.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2004.
  * (70) T. Eguchi and H. Kawai, “Reduction of Dynamical Degrees of Freedom in the Large N Gauge Theory,” Phys. Rev. Lett. 48 (1982) 1063.