# Notes on Confinement on $\mathbf{R^{3}\times S^{1}}$: From Yang-Mills,
super-Yang-Mills, and QCD(adj) to QCD(F)111An expanded and updated version of
lectures originally given at the 22nd W. E. Heraues “Saalburg” Summer School
in Wolfersdorf, Germany, in September 2016. Prepared for the 2021 Special
Issue of the journal “Symmetry” on “New Applications of Symmetry in Lattice
Field Theory.”

Erich Poppitz Department of Physics, University of Toronto
60 St George St, Toronto, ON M5S 1A7, Canada
<EMAIL_ADDRESS>
###### Abstract

This is a pedagogical introduction to the physics of confinement on
$R^{3}\times S^{1}$, using $SU(2)$ Yang-Mills with massive or massless adjoint
fermions as the prime example; at the end, we also add fundamental flavours.
The small-$S^{1}$ limit is remarkable, allowing for controlled semiclassical
determination of the nonperturbative physics in these, mostly non-
supersymmetric, theories. We begin by reviewing the Polyakov confinement
mechanism on $R^{3}$. Moving on to $R^{3}\times S^{1}$, we show how
introducing adjoint fermions stabilizes center symmetry, leading to
abelianization and semiclassical calculability. We explain how monopole-
instantons and twisted monopole-instantons arise. We describe the role of
various novel topological excitations in extending Polyakov’s confinement to
the locally four-dimensional case, discuss the nature of the confining string,
and the $\theta$-angle dependence. We study the global symmetry realization
and, when available, present evidence for the absence of phase transitions as
a function of the $S^{1}$ size. As our aim is not to cover all work on the
subject, but to prepare the interested reader for its study, we also include
brief descriptions of topics not covered in detail: the necessity for analytic
continuation of path integrals, the study of more general theories, and the ’t
Hooft anomalies involving higher-form symmetries.

###### Contents

  1. 1 Introduction.
    1. 1.1 What are these lectures about?
    2. 1.2 Why study small-$L$ theories? In lieu of conclusion.
    3. 1.3 Philosophy and a reader’s guide.
  2. 2 Flashback to the 1970’s: Polyakov confinement on $\mathbf{R^{3}}$.
    1. 2.1 Perturbative analysis of the IR theory.
    2. 2.2 Finite action monopole-instantons in the Polyakov model.
    3. 2.3 Monopole-instantons and the IR: I. first pass.
    4. 2.4 Monopole-instantons and the IR: II. duality, ’t Hooft vertices, and monopole operators.
    5. 2.5 Monopole-instantons and the IR: III. dilute gas and mass gap.
    6. 2.6 Monopole-instantons and the IR: IV. confinement and the string tension.
  3. 3 From $\mathbf{R^{3}}$ to $\mathbf{R^{3}\times S^{1}}$: generalities.
    1. 3.1 Holonomy, Polyakov loop, center symmetry, and the Weyl chamber.
    2. 3.2 Spectrum near center-symmetry. When does small-${L}$ imply weak coupling?
    3. 3.3 $M$ and $KK$ monopole-instantons.
    4. 3.4 Adjoint fermions, “GPY” potential, and center stability.
      1. 3.4.1 The adjoint spectrum on the Weyl chamber.
      2. 3.4.2 The one-loop GPY potential via supersymmetry.
      3. 3.4.3 Digression: $Z_{2}^{(1)}$-breaking and deconfinement in high-$T$ pure-YM theory.
      4. 3.4.4 Stabilizing center symmetry via massless or massive adjoints.
  4. 4 Back to the 2010’s: the properties of dYM.
    1. 4.1 Mass gap, string tension, their $\theta$-dependence, and “adiabatic continuity.”
    2. 4.2 $\theta=\pi$.
      1. 4.2.1 $\theta=\pi$: spontaneous breaking of parity.
      2. 4.2.2 $\theta=\pi$: “double-string” confinement and deconfinement on domain walls.
      3. 4.2.3 $\theta=\pi$: a first encounter with magnetic bions.
    3. 4.3 Qualitative picture of the small-$L$, finite-$T$ deconfinement transition on $R^{2}\times S^{1}\times S^{1}_{\beta}$.
  5. 5 QCD(adj).
    1. 5.1 Discrete chiral symmetry and its action in the EFT.
    2. 5.2 Symmetry constraints on the QCD(adj) EFT.
    3. 5.3 The infrared dynamics of QCD(adj) on the circle.
      1. 5.3.1 The composite nature of magnetic bions and a picture of the QCD(adj) vacuum.
      2. 5.3.2 Symmetry realization, domain walls, and confinement.
      3. 5.3.3 Possible (new) large-$L$ phases.
  6. 6 SYM.
    1. 6.1 Fields, symmetries, ’t Hooft vertices, and the allowed terms in the EFT.
    2. 6.2 Vacua, domain walls, “double-string” confinement, and liberation of quarks on domain walls.
    3. 6.3 Neutral bions and the need for analytic continuation of path integrals.
    4. 6.4 Neutral bions and the pure-YM thermal deconfinement transition: the continuity conjecture.
  7. 7 QCD(F) and “colour-flavour-center” symmetry.
    1. 7.1 Colour-flavour-center symmetry.
    2. 7.2 Chiral symmetry breaking on $R^{4}$ and $R^{3}\times S^{1}$.
    3. 7.3 Continuity between the large-$L$ and small-$L$ symmetry realization.
  8. 8 A quick guide to the literature on other theories.
    1. 8.1 dYM, QCD(adj)/SYM, QCD(F) with gauge group $SU(N)$ and $N>2$.
    2. 8.2 Other matter representations with $SU(N)$ gauge group.
    3. 8.3 Other gauge groups.
  9. A Notations and conventions.
  10. B The massless adjoint fermion contribution to “GPY” potential.
  11. C The massive adjoint fermion contribution to “GPY” potential.
  12. D A flavour of the $R^{3}\times S^{1}$ index theorem: $SU(2)$ spin-$j$ representations.

##  1 Introduction.

I have overheard graduate students, after having taken a standard quantum
field theory course, say that “confinement occurs because the beta function is
negative and the coupling becomes strong at long distances.” Loosely, I
translate this to “it’s complicated and I won’t think about it.” That the
students’ explanation is insufficient is underscored by the fact that we now
know, due to insight from supersymmetry [1], of gauge theories where the
coupling becomes strong, but which become nontrivial conformal field theories
at long distances. Conversely, while confinement is a nonperturbative
phenomenon, as evidenced by the framework described in these notes, it does
not always require strong coupling.

Nonetheless, the students’ attitude is not unreasonable: in real-world QCD the
long-distance physics of chiral symmetry breaking and the emergence of hadrons
occurs at strong coupling, making it difficult to handle analytically. Hence,
many are happy to leave222While others stick to the already mentioned
supersymmetric world (a full disclosure requires us to state that these notes
will also not remain immune to the charms of supersymmetry). the study of
strong-coupling phenomena to experiments, performed either in the lab or on
the computer. Indeed, the numerical approach to lattice QCD has been very
successful in relating the short-distance description in terms of quark and
gluon fields to the hadron physics emerging at long distances.

In summary,333The quotation marks indicate the need to more carefully define
what we mean. See Sections 2.6, 3.1, 3.4.3. “confinement” is 45+ years old
news and we have gotten used to it. It is believed to be a property of pure
Yang-Mills theory, but we have no proof.444For extra motivation, see the Clay
Institute website https://www.claymath.org/millennium-problems. Even a more
modest goal, a physicist’s analytical understanding in continuum
asymptotically-free pure Yang-Mills theory on $R^{4}$ is lacking. We do have
analytical understanding via the strong-coupling expansion on the lattice, but
this is far from the continuum limit. There is also the overwhelming numerical
evidence from lattice simulations.

At this point, let me make a disclaimer and a recommendation. The disclaimer
is that I can not possibly review all existing analytical—theoretically
controlled or otherwise—and numerical approaches to confinement. The
recommendation is, for an overview of the existing ideas about confinement, to
consult the comprehensive and refreshingly (self-) critical monograph [2] (a
minor warning is that its relatively small size is both a blessing and a
curse).

After this preamble, we now turn to the topic of these notes.

###  1.1 What are these lectures about?

These lectures are about an approach to the study of confinement that emerged
within the past decade or so. This is an analytical approach to confinement,
within asymptotically free QFT in four dimensions (thus, not using AdS/CFT and
other string-inspired tools [3]) that is under theoretical control. In the
case at hand, a weak-coupling semiclassical expansion using objects defined in
the UV theory is valid and the physics is weakly coupled at all scales, all
the way from the UV to the IR. This makes the $R^{3}\times S^{1}$ setup
distinct from the few analytical approaches on $R^{4}$ that do not involve
uncontrolled approximations, notably Seiberg-Witten theory [4]. This is Yang-
Mills theory with extended ${\cal{N}}=2$ supersymmetry, with a soft mass term
preserving minimal ${\cal{N}}=1$ supersymmetry. Here, owing to supersymmetric
dualities, the IR physics has a weakly coupled description and confinement can
be shown to be due to the condensation of monopole or dyon particles, in a
kind of dual Higgs mechanism. Other examples, also in theories with (extended)
supersymmetry, describe a type of confinement dual to the confinement of
electric charges: the confinement of monopoles via nonabelian strings, see
e.g. [5] and references therein.

The scope of these lectures is as follows: we study the $R^{3}\times S^{1}$
approach to nonperturbative physics on the example of four-dimensional $SU(2)$
nonabelian gauge theory with a number of Weyl fermions, denoted by $n_{f}$, in
the adjoint (or vector) representation of the gauge group. The theory is
asymptotically free for $n_{f}\leq 5$. The fermions can be taken massless, or
be given gauge invariant Majorana masses. The masses of the different flavours
can be taken different, but for our discussions here we shall assume them to
be of the same order and denote their overall scale by $m$. As mentioned
before, the class of theories with calculable nonperturbative dynamics is
larger, but we stick to $SU(2)$ for pedagogical reasons.555While there are
many interesting observations to be made upon replacing $SU(2)$ with $SU(N)$,
there is already more than enough material to cover; thus, introducing more
group theory is left for future self-study.

When formulated on $R^{4}$, analytical approaches to the nonperturbative
physics determining the vacuum structure and symmetry realization are not
available, due to the strong-coupling infrared problem. Here, we shall study
the above class of theories, but formulated on $R^{3}\times S^{1}$, with
periodic boundary conditions for the fermions. Thus, our $S^{1}$ is not a
thermal circle (which would require anti-periodic boundary conditions for the
fermions) but a compact spatial direction666A purist would then insist that we
refer to $R^{1,2}\times S^{1}$ instead, but since most of our studies will be
Euclidean, for brevity we stick with $R^{3}\times S^{1}$. of circumference
$L$. As we shall explain, calculability is ensured by taking the size of the
$S^{1}$ circle to be smaller than the inverse of the strong coupling scale of
the four-dimensional theory $\Lambda$, often called the “confinement scale.”

Before we continue, we note that the idea of the “femto-universe,” where Yang-
Mills theory is considered in a small volume, was first put forward by Bjorken
long ago [6].777Incidentally, Bjorken is also the one rumoured to have coined
the phrase “voodoo QCD,” presumably to characterize approaches to
nonperturbative physics whose validity is not a priori clear and is hard to
justify (except for agreement with a set of data, often judiciously chosen).
Jokes aside, while modelling the nonperturbative dynamics has its raison
d’être, our emphasis here is on studying gauge theories from first principles
and without uncontrolled approximations, hence we stay far from such
approaches.  He envisaged, however, taking all three directions of space
smaller than the confinement scale, of order $\sim 1$ Fermi, hence the name.
Significant effort has gone into studying the physics of Yang-Mills theories
on small spatial three-tori since [7], see the review [8]. It turns out that
formulating the perturbative and semiclassical expansion on a small $T^{3}$ is
quite complex and difficult to handle. In addition there is a center-symmetry
breaking phase transition separating the small-$T^{3}$ theory from the
infinite volume one. However, it also turns out that taking two of the
dimensions of the $T^{3}$ be infinite, thus considering the limit $R^{1}\times
T^{3}\rightarrow R^{1,2}\times S^{1}$, brings, in a large class of theories,
great advantages with respect to calculability, at least to leading order in
the semiclassical expansion.888Due to important insight from the 1990’s, see
[9, 10] and Section 3. A center-symmetry breaking phase transition upon
compactification can also be avoided.

It was first realized by Ünsal [11, 12] that the nonperturbative physics of
these theories at $L\Lambda\ll\pi$ can be studied in a weak-coupling
semiclassical expansion. One can show that a nonperturbative mass gap for the
gauge fluctuations is generated, a confining string can be seen to form when
charged probes are inserted in the vacuum, and, in the theories with chiral
symmetries, one can study their realization in the ground state and show their
spontaneous breakdown. In addition, a center-symmetry breaking phase
transition, typically expected after compactification, can be avoided and one
can argue for a smooth connection to the $R^{4}$ theory of ultimate interest.

The space of $SU(2)$ theories with adjoint fermions, defined by the parameters
$\Lambda,n_{f},m$, and the $S^{1}$ size $L$, can be roughly split into three
classes, each with distinct calculable dynamics at $L\Lambda\ll\pi$. We
enumerate them below in the order we study them in these lectures. We
introduce our jargon (dYM, QCD(adj), SYM) and also briefly advertise their
main properties that shall be explained below:

  1. 1.

dYM, or deformed Yang-Mills theory [13, 14]: for our purpose, this theory is
defined in a UV complete manner as Yang-Mills theory with two or more massive
adjoint fermions, whose mass is taken $m\sim 1/L$. The fermions decouple from
the physics at energy scales $\ll 1/L$, but leave an important imprint: center
stability and abelianization, ensuring calculability of the long-distance
physics. Confinement and the $\theta$-angle dependence can be analytically
studied. Notice that dYM is in the universality class of pure Yang-Mills
theory. It is believed to be continuously (i.e. without phase transition)
connected to the $R^{4}$ pure Yang-Mills theory. The agreement with lattice
data, particularly regarding topological properties is quite remarkable.

  2. 2.

QCD(adj), or adjoint QCD, is the theory with $2\leq n_{f}\leq 5$ massless
adjoint Weyl fermions. The long distance physics of confinement and chiral
symmetry breaking at $L\Lambda\ll\pi$ is calculable [11]. The study of this
theory highlights the importance of novel topological excitations in the
dynamics of confinement and chiral symmetry breaking, the so-called “magnetic
bions.” The structure of confining strings is markedly different from that in
dYM, reflecting certain recently found “generalized” ’t Hooft
anomalies.999“Generalized ’t Hooft anomalies” is a topic that we can not cover
in any detail here, see [15, 16, 17].

  3. 3.

SYM, or supersymmetric-Yang-Mills, is the same as QCD(adj), but with
$n_{f}=1$. This theory stands out because it automatically has ${\cal{N}}=1$
supersymmetry (for $m=0$ only). Thus, it can be studied using the powerful
tools of holomorphy introduced by Seiberg [1]. In addition, just like
QCD(adj), it also becomes semiclassical at small $L$, allowing for a
calculation of the mass gap and confinement properties, not accessible to the
“power of holomorphy.” It turns out that the structure of confining strings in
SYM is like that in QCD(adj), for similar anomaly-related reasons. Most
tantalizingly, in the semiclassical limit one finds a novel kind of
topological excitation [18, 19, 20, 21] the so-called “neutral bions,”
ensuring center-stability and abelianization.101010These topological
excitations highlight the need for analytic continuation of path integrals and
are, ultimately, relevant for the idea of “resurgence,” also outside the topic
of these lectures (see Section 6.3 for motivation and brief discussion).

Obviously, by varying the mass parameters of the fermions (i.e. taking some
masses to be $\gg 1/L$ allows for the decoupling of flavours) one can arrange
for interesting renormalization group flows between the above theories. In
some cases, one can also make conjectures about the nature of the thermal
deconfinement transition in pure Yang-Mills theory which can be confronted by
lattice data (we shall consider an example in these notes).

Summary of 1.1:

The answer to the question posed in the title of this section is: we shall
explain in some detail how the nonperturbative properties listed above arise
in the $L\Lambda\ll\pi$ limit.

###  1.2 Why study small-$L$ theories? In lieu of conclusion.

Skeptical students (and colleagues) often ask: “You are going to tell us how
to study gauge theories on a small circle. To boot, you study theories with
unphysical—adjoint, not fundamental as in the standard model—fermion
representations. This is definitely not the real world. Why bother?” This is a
fair question: if your interest is mainly in a calculation of hadron spectra
that can be directly compared with experiment, stop reading; if you are
interested in the inner workings of QFT, stay on. As my goal is not to (over-)
sell you anything, I will simply tell you why I think this topic is
interesting.

First, as we shall see, the stories I will tell about confinement, chiral
symmetry breaking, and the thermal deconfinement transition are rather
elegant. It is a rare luxury to be able to make statements about the
nonperturbative phase structure of a locally four dimensional quantum field
theory. To a theoretical physicist, this alone is quite satisfying. It makes
the study intrinsically worthwhile and fun.

Second, it can be expected that, while honestly studying a theoretically
controlled regime, one may encounter surprising new features that are more
generally valid. The recent resurgence of interest, reviewed in [22], in
“resurgence in QFT” (studying the nature of the divergent perturbative series
in QFT and their resummation) did, in fact, arise from these small-$L$
studies. This was due to, among others, the peculiar nature of the magnetic
and neutral bion topological molecules. Other unusual features observed in the
calculable regime, in addition to the many novel and strange topological
excitations (Sections 5.3.1 and 6.3), include the appearance of doubly-
exponential nonperturbative effects (Sections 4.1 and 6.2) and the emergence
of latticized dimensions in the abelian large-$N$ limit (Section 8 and [23]).

Third, the theoretically-controlled study of the symmetry realization at small
$L$ has led to the conjectured existence of novel possible phases of various
theories on $R^{4}$ (Sections 5.3.3 and 8). Admittedly, one needs dedicated
numerical studies to confirm or refute these conjectured phases. Further, we
can also use SYM with soft-breaking mass on $R^{3}\times S^{1}$ to make
conjectures about the nature of the thermal phase transition in pure YM
theory. These are borne out by available numerical simulations (Section 6.4).

Fourth, the small-$L$ theories offer an interesting arena to study the
realization, at various energy scales of the recently discovered ’t Hooft
anomalies involving traditional, or “0-form,” global symmetries as well as
higher-form symmetries; their implications are discussed in Sections 4.2.2,
5.3.2, 6.2, and 8. One’s hope, then, is that this will lead to better
understanding of these anomalies and their implications.

Finally, I will also mention that theories with fundamental fermions have been
incorporated in an interesting way. They exhibit various nonperturbative
properties of real QCD in a calculable setup. At small $L$, chiral symmetry is
broken by the expectation value of a monopole operator and one can argue that
a chiral phase transition between the small and large $L$ theories is absent.
We explain this in detail, for $SU(2)$ QCD(F) (Section 7).

Summary of 1.2:

The calculable nonperturbative dynamics on $R^{3}\times S^{1}$ offers a rare
theoretical opportunity to analytically study nonperturbative phenomena in 4d
gauge theories. My answer to the question posed in the title of this Section
is that this alone makes these explorations worthwhile, as they extend our
understanding of the nonperturbative properties of quantum field theories in
ways that are not always obvious from the start.

In lieu of conclusion: Our hope is that these notes give the necessary
background, collected all in one place, to help the interested reader through
the literature on $R^{3}\times S^{1}$ compactifications. The main emphasis is
on introducing the ideas and techniques used to study the small-$L$ calculable
regime in a range of different theories, focusing on IR energy scales $\mu\ll
1/L$ and exhibiting the physics of confinement and (chiral) symmetry breaking.
On a few occasions,111111As in Sections 7.2 and 4.1. We stress that despite
the fact that the small-$L$ theory is weakly coupled at all scales, there are
situations (as mentioned in Section 4.2.2) where accounting for this
backreaction is a nontrivial open problem. the backreaction of the IR physics
on the UV modes of mass $1/L$ will be also mentioned. As the reader will no
doubt notice, the continuity of the symmetry realization towards large $L$ is
largely conjectural, with evidence based on comparisons with lattice data or
on other expectations, such as consistency with anomalies. Notably a
theoretical proof of (the absence of) large-$L$/small-$L$ continuity in most
of the theories we discuss, apart from SYM, is not known. Clearly, there is
room to advance our understanding.

###  1.3 Philosophy and a reader’s guide.

These notes assume some background knowledge. This includes the basics of non-
abelian gauge theory and asymptotic freedom as well as some familiarity with
’t Hooft-Polyakov monopoles, chiral anomalies, instantons, and the dilute
instanton gas approximation, in quantum mechanics and on $R^{4}$ (references
covering some of this material in more depth are provided throughout). A few
general remarks regarding the philosophy behind these notes are due:

  1. 1.

As the notes are rather long, every subsection ends with a paragraph
summarizing the main results (highlighted in red). This should help the
returning reader while allowing those familiar with the subject of a given
Section to quickly review its content.

  2. 2.

We have tried to balance hand-waving explanations and careful derivations,
often leaning towards the former. We feel that keeping in mind the order of
magnitude and the leading parametric dependence of the physical quantities is
more important for a qualitative understanding than the precise numerical
factors.121212We did, however, make every effort to have the correct numerical
coefficients when they really matter—in discussions of charge quantization,
monodromies, and other topological features. This attitude will be pervasive
on many occasions in these notes and various factors we omit can be found in
the literature.

  3. 3.

Throughout the notes, some straightforward technical derivations are relegated
to exercises, intended to improve the reader’s appreciation of the topics.
These are especially recommended to those encountering them for the first
time. However, the notes can be read without solving the exercises, by simply
understanding (and trusting) the statements they make.

To help the readers—who might have vastly different backgrounds and
interests—through these long notes, we now include some guidance of how one
may go through the various Sections. We stress that to understand the
post-2010 developments, familiarity with background discussed in two of the
eight Sections of these notes, Sect. 2 and 3, is an absolute necessity. Due to
their importance they take about half of the space.

The first is Section 2, where we discuss in detail Polyakov’s confinement on
$R^{3}$, known since the 1970’s.

The second is Section 3, devoted to notions specific to $R^{3}\times S^{1}$.
These include the holonomy, center symmetry, its relation to
confinement/deconfinement, as well as the holonomy (or “GPY”) potential,
familiar from the 1980’s. These are discussed in Sections 3.1 and 3.2. A newer
crucial development on $R^{3}\times S^{1}$ is the “dissociation” of an
instanton into its monopole-instanton constituents, the “M” and “KK” monopole-
instantons (for $SU(2)$), discovered in the 1990’s and explained in Section
3.3. The final crucial set of ideas, from the 2000’s, is the perturbative
center-symmetry stabilization due to massive or massless adjoints, the subject
of Section 3.4 and Appendices B and C.

Readers familiar with the subjects of Sections 2 and 3 can proceed directly to
the Sections devoted to studying the nonperturbative dynamics of the classes
of theories they may be interested in:

Section 4 discusses deformed Yang-Mills theory, adjoint QCD is the subject of
Section 5, and super-Yang-Mills is in Section 6. The many topics discussed for
each class of theories can be found in the table of contents.

Section 7 is devoted to a detailed description of $SU(2)$ QCD with fundamental
quarks in the context of colour-flavour-center symmetric compactifications.

Section 8 contains much briefer descriptions and references involving all
classes of theories discussed here, but for $SU(N)$ gauge groups with $N>2$,
as well as other classes of theories and gauge groups.

##  2 Flashback to the 1970’s: Polyakov confinement on $\mathbf{R^{3}}$.

We begin our journey towards confinement on $R^{3}\times S^{1}$ by studying a
theory in $R^{3}$. This is the so-called “Polyakov model” [24, 25]. More
recent treatments can be found in Witten’s lectures in vol. 2 of [26] and, for
example, in the textbooks by Shifman [27] and Banks [28]. As this is not
material taught in standard QFT courses, we include an extended discussion,
intended to be self-contained. (The familiar reader can skip this Section and
move to Section 3 discussing $R^{3}\times S^{1}$.)

The Polyakov model is an $SU(2)$ gauge theory with a real adjoint scalar field
in three dimensional spacetime.131313On $R^{4}$, this theory is known as the
bosonic sector of the “Georgi-Glashow model,” a pre-cursor of modern electro-
weak theory, which lacks the $Z$-boson mediated neutral currents. The gauge
field is $A_{\mu}^{a}$.141414See Appendix A for a summary of our notation and
a few warnings about possible confusions.  The real scalar field in the
adjoint representation of $SU(2)$ shall be labeled $A_{4}^{a}$, with some
hindsight aimed towards our future $R^{3}\times S^{1}$ study. The theory has
the Euclidean Lagrangian:

$L={1\over 4g_{3}^{2}}F_{\mu\nu}^{a}F^{\mu\nu\;a}+{1\over
2g_{3}^{2}}(D_{\mu}A_{4})^{a}(D^{\mu}A_{4})^{a}+{\lambda\over
g_{3}^{2}}(A_{4}^{a}A_{4}^{a}-v^{2})^{2}\leavevmode\nobreak\ ,$ (1)

where the 3d gauge coupling $g_{3}^{2}$ has dimension of mass and the gauge
field $A_{\mu}^{a}$ and scalar $A_{4}^{a}$ both have mass dimension one. We
have normalized the entire Lagrangian so that the action $S=\int\limits
d^{3}xL$ has an overall factor $1/g_{3}^{2}$. As $g_{3}^{2}$ is dimensionful,
the role of $\hbar$, the semiclassical expansion parameter is played by a
ratio $E/g_{3}^{2}$, where $E$ is a relevant energy scale, the nature of which
shall be revealed below. Finally, $\lambda$ is a dimensionless coupling, which
shall be assumed to be ${\cal{O}}(1)$ or smaller (also see further below).

###  2.1 Perturbative analysis of the IR theory.

The potential for the adjoint (triplet, or vector) scalar $A_{4}^{a}$ is
similar to that found for the Higgs field in the standard model. To minimize
the energy, we let $A_{4}$ have an expectation value (vev), which we take
along the 3rd “isospin” direction:

$\langle A_{4}^{a}\rangle=\delta^{a3}v\leavevmode\nobreak\ .$ (2)

Expanding around the vev for $A_{4}$ it is easy to see that the gauge symmetry
is broken, $SU(2)\rightarrow U(1)$. For the choice of vev we have made, the
massless gauge boson is $A_{\mu}^{3}$, while $A_{\mu}^{1}\pm iA_{\mu}^{2}$ are
massive $W^{\pm}$ bosons. Their mass, as can be seen from (1) is of order $v$.
The $A_{4}^{1}$ and $A_{4}^{2}$ components of the triplet scalar are “eaten”
by the massive gauge bosons, while the fluctuation of $A_{4}^{3}$ around the
vev (2), the radial component of the higgs field, has mass which depends on
the parameter $\lambda$ and is of order $\sqrt{\lambda}v$. For $\lambda\sim
1$, this is of the same order as the mass of the $W$-bosons, while taking
$\lambda\ll 1$ makes for a hierarchy in the massive spectrum.

Next, observe that in this model, there are two dimensionful parameters: the
scale of the vev (2), $v$, and the scale set by the 3d gauge coupling,
$g_{3}^{2}$. In our classical analysis, the scale $v$ sets the mass of the
heavy $W$-bosons and radial higgs mode $A_{4}^{3}$. We shall call the vev $v$
the scale of “abelianization,” i.e. the scale below which the nonabelian
degrees of freedom decouple and the only relevant (here: massless) degree of
freedom is that of the $U(1)$ “photon” $A_{\mu}^{3}$.

If we had set the $v^{2}$ to zero (or taken it negative), we wouldn’t have had
a “Mexican hat” potential for the triplet scalar and the theory would remain
nonabelian at all scales. What, then, is the IR dynamics expected of such an
$SU(2)$ theory (+ adjoint scalar) in 3d? Based solely on dimensional analysis,
one expects that at scales of order $g_{3}^{2}$, the 3d theory becomes
strongly coupled. Such behaviour is typical in lower dimensional theories,
where the couplings have positive mass dimension. These represent relevant
parameters whose importance grows in the IR. Formally, as in any theory with
dimensionful coupling constant, one can define a dimensionless coupling using
an appropriate power of the relevant energy scale $\mu$. In our case, as
$g_{3}^{2}$ has dimension of mass (and is, by standard power-counting, a
relevant coupling) the dimensionless combination is $g_{3}^{2}\over\mu$,
showing that this coupling grows in the IR and becomes large when $\mu\sim
g_{3}^{2}$. Those not familiar with these sort of arguments should follow the
more explicit discussion between eqns. (3) and (4) below.

However, if the scale $v$ is taken to be $v\gg g_{3}^{2}$, the abelianization
of the theory stops the running of the coupling towards large values in the
IR. At energies less than $v$ the theory is free, as there is only the
$A_{\mu}^{3}$ photon and there are no $U(1)$-charged particles that are
light.151515Notice that this is crucial in 3d where a $U(1)$ theory with light
charged particles becomes strongly coupled in the IR. The point we shall make
more explicit below is that our analysis of the spectrum above is sensible
only in the $v\gg g_{3}^{2}$ limit. Thus, the dimensionless expansion
parameter in our theory is expected to be ${g_{3}^{2}\over v}\ll 1$. Its
smallness guarantees the validity of the (essentially classical) analysis
above.

Figure 1: The one-loop heavy $W$-boson contribution to the photon kinetic
term. From (1), all vertices come with a factor ${1\over g_{3}^{2}}$ and the
$W$-boson propagator is $\sim{g_{3}^{2}\over{k^{2}+v^{2}}}$. Thus, the loop
integral scales as $\int{d^{3}k\over(k^{2}+v^{2})^{2}}\sim{1\over v}$,
producing the second term in $L_{eff}$ of (3), with a calculable coefficient
$C$. (For this estimate, the momenta at the interaction vertices can be taken
to act on the external $A_{\mu}^{3}$ to produce the field strength in (3)).

Let us elaborate on the above two paragraphs. To this end, we write an
effective Lagrangian governing the physics at $\mu\ll{\rm
min}(v,\sqrt{\lambda}v)$. We claim that it has the schematic form:

$L_{eff}={1\over 4g_{3}^{2}}(F_{\mu\nu}^{3})^{2}\left[1+C{g_{3}^{2}\over
v}\right]+\ldots\leavevmode\nobreak\ ,$ (3)

where $C$ is a numerical coefficient. The overall term multiplying the square
brackets is self-explanatory: we simply kept the kinetic term for the massless
photon $A_{\mu}^{3}$ from (1) and ignored the terms involving massive fields
(this is equivalent to the “tree-level” integrating out the heavy fields;
here, it amounts to crossing them out). The second term in the brackets can be
seen as the result of the calculation of the one-loop graph shown on Fig. 1
(naturally, there are higher loop corrections not shown), and the dots in (3)
denote higher-dimensional terms suppressed by powers of the $W$-boson mass $v$
(e.g. $\sim(F_{\mu\nu}^{3})^{3}$, not showing the Lorentz index contractions),
which are irrelevant at $\mu\ll v$.161616This is similar to the Heisenberg-
Euler lagrangian in QED describing the photon self-interactions at energies
below the mass of the electron. The precise calculation of the second term in
the brackets requires some care. However, simple dimensional analysis suffices
to guess the parametric scaling of the operator given in the term in $L_{eff}$
multiplied by $C$. The normalization of our lagrangian (1) determines the
scaling of vertices and propagators and thus the final scaling of the one-loop
result, as explained in the caption of Fig. 1.

To conclude, the dimensional analysis explained in Figure 1 determines the
form of the term in square brackets in (3). It tells us that the $U(1)$ gauge
coupling in the effective theory of the massless modes is of the form

${1\over g_{eff}^{2}}={1\over g_{3}^{2}}+{C\over v}+\ldots\leavevmode\nobreak\
,$ (4)

given by the sum of a tree-level term and the loop corrections. It is clear
from (4) that perturbation theory breaks down whenever the second term becomes
bigger than the first,171717The precise calculation reveals the true
behaviour: the sign of $C$ is negative. Taken at face value, it implies that
the coupling of the massless photon blows up at a finite value of $v$,
indicating a breakdown of the abelian effective field theory. This is another
way to see the relevant nature of the gauge coupling in the 3d Yang-Mills
theory. See [29] for a calculation in a supersymmetric 3d theory. For a
nonsupersymmetric theory, this behaviour can be inferred from the $R^{3}\times
S^{1}$ calculation in QCD(adj) [30], after throwing out the contributions of
adjoint fermions and Kaluza-Klein modes on the circle. i.e. when $v\ll
g_{3}^{2}$. Conversely, as advertised earlier, when $v\gg g_{3}^{2}$ the one-
loop correction is small and the semiclassical (for now: perturbative)
approximation works, at the usual physicist level of rigour. As already
declared, this is the limit we shall be exploring to study the physics of the
Polyakov model semiclassically.

Summary of 2.1:

The upshot of our analysis here is that the perturbative IR physics of the
Polyakov model (1), taken with $v\gg g_{3}^{2}$, is rather boring: there is a
free massless photon, which has irrelevant self interactions due to the
“$\ldots$” terms in (3). However, as we shall see in the following sections,
nonperturbative effects due to (1), still calculable in the semiclassical
limit, completely change the IR behaviour. A mass gap for the perturbatively
free IR $U(1)$ theory is generated nonperturbatively, giving rise to
confinement of probe electric charges.

###  2.2 Finite action monopole-instantons in the Polyakov model.

The striking fact about the Polyakov model is that the simple free-photon
lagrangian (3) does not give a correct account of the IR physics.
Nonperturbative effects, not accounted for by any finite number of loop
diagrams, like that on Fig. 1, completely change the long distance physics.
The goal of this section is to discuss the objects responsible for this change
of IR behaviour: the “monopole-instantons” of the Polyakov model (1). It is
their proliferation in the vacuum that ultimately causes confinement of
fundamental charges.

The monopole-instantons are finite Euclidean action solutions of the equations
of motion of the Polyakov model (1), akin to the famous Belavin-Polyakov-
Schwarz-Tyupkin (BPST) instantons in $R^{4}$. In order to exhibit them in a
straightforward manner, we shall work in the $\lambda=0$ limit, where
$A_{4}^{a}$ is a scalar field without a potential and thus $v$ is arbitrary.
Studying this limit is, strictly speaking, not necessary in the purely 3d
theory. However, in our $R^{3}\times S^{1}$ study, the physics of the
compactified 4d theory will force us into a regime where the scalar potential
is small, i.e. $\lambda\ll 1$. The $\lambda=0$ limit is known as the
“Bogomolny-Prasad-Sommerfield,” or BPS, limit, known to simplify the study of
monopole-instantons.181818As stated earlier, some familiarity of the reader
with ’t Hooft-Polyakov monopoles is assumed here and our discussion may appear
terse. For a review of BPS monopoles, see Harvey’s lectures [31] (our
monopole-instantons are related to the finite-energy solitonic monopole
solutions on $R^{1,3}$ discussed there by forgetting the time direction). A
pedagogical reference that also goes beyond the BPS limit is E. Weinberg’s
book [32]. (As an aside, this limit is natural in supersymmetric theories,
where a vanishing potential, $\lambda=0$, is perturbatively stable.)

In the $\lambda=0$ limit, we can simplify matters further if we write (1) in a
“4d” form as follows. Let us introduce indices $M,N=1,2,3,4$ and extend the
Euclidean metric to the usual 4d one. Then, observe that

$L={1\over 4g_{3}^{2}}F_{\mu\nu}^{a}F^{\mu\nu\;a}+{1\over
2g_{3}^{2}}(D_{\mu}A_{4})^{a}(D^{\mu}A_{4})^{a}={1\over
4g_{3}^{2}}F_{MN}^{a}F^{MN\;a}\leavevmode\nobreak\ ,$ (5)

where
$F_{MN}^{a}=\partial_{M}A_{N}^{a}-\partial_{N}A_{M}^{a}+i([A_{M},A_{N}])^{a}$,
and $F_{\mu 4}^{a}=(D_{\mu}A_{4})^{a}$, i.e. $\partial_{4}\equiv 0$, the
dependence on the fictitious (for now) 4th dimension is neglected. Again, I
stress that this 4d terminology is simply convenient for the study of the
$R^{3}$ theory, where no $\partial_{4}$ appears, and will only become
indispensable once we transition to $R^{3}\times S^{1}$. Further, let us also
introduce the components of the field-strength tensor:

$B_{\mu}^{a}={1\over
2}\epsilon_{\mu\nu\lambda}F^{\nu\lambda\;a}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{\mu}^{a}=(D_{\mu}A_{4})^{a}=F_{\mu 4}^{a}\leavevmode\nobreak\ ,$ (6)

and call these fields “magnetic” and “electric,” respectively
($\epsilon_{123}=1$). Notice that the names we attach to the above $B$ and $E$
are a convenient choice of words and should not be taken literally, as these
are Euclidean $R^{3}$ fields. If we go to Minkowski space, $R^{1,2}$, by say,
declaring $x^{1}$ to be the Euclidean version of time, $B_{1}$ will represent
the true magnetic field (a pseudoscalar $F_{23}$ in 3d) while $B_{2}$ and
$B_{3}$ are actually electric fields, related to $F_{12}$ and $F_{13}$ by (6).
On the other hand, $E_{\mu}$ is simply the covariant derivative of a
scalar.191919There exists confusing nomenclature in the literature related to
taking the “electric” and “magnetic” labels attached to $E$ and $B$ seriously.
The reason for why these names stuck will become clear shortly. It would be
nice to banish them, but they have by now been deeply ingrained, so we use
them but remember the context.  Now, using (6) we can rewrite (5) (for
brevity, we do not distinguish upper and lower indices, all are assumed to be
lowered/raised by the unit metric) as follows

$L={1\over 2g_{3}^{2}}\left[(B_{\mu}^{a})^{2}+(E_{\mu}^{a})^{2}\right]={1\over
2g_{3}^{2}}\left[(E_{\mu}^{a}\mp B_{\mu}^{a})^{2}\pm
2E_{\mu}^{a}B_{\mu}^{a}\right]\leavevmode\nobreak\ .$ (7)

The above equation, due to the positivity of $(E_{\mu}^{a}\pm
B_{\mu}^{a})^{2}$, implies that the (positive definite) Euclidean action obeys
an inequality known as the “BPS bound”

$S=\int d^{3}xL\geq\pm{1\over g_{3}^{2}}\int
d^{3}xE_{\mu}^{a}B_{\mu}^{a}\equiv S_{BPS}\leavevmode\nobreak\ .$ (8)

What is interesting about these manipulations? Suppose we are looking for
finite Euclidean action solutions of the equations of motion of (5), i.e. for
instantons. Finiteness of the action requires that $E^{2}$ and $B^{2}$, and
hence, $(E\pm B)^{2}$ and $E\cdot B$ are all integrable on $R^{3}$. Then, (8)
implies that the action will reach its minimum value $S_{BPS}={1\over
g_{3}^{2}}\int d^{3}x|E_{\mu}^{a}B_{\mu}^{a}|$ whenever $E=\pm B$. Solutions
of the equations of motion of (5) which obey $E_{\mu}^{a}=B_{\mu}^{a}$ are
called “self-dual,” while the ones where $E_{\mu}^{a}=-B_{\mu}^{a}$ are the
“anti-self-dual” ones. Further, we notice that $E_{\mu}^{a}B_{\mu}^{a}$ is a
total divergence, hence (taking the positive sign for definiteness and
denoting by $d^{2}s^{\mu}$ the surface element)

$S_{BPS}={1\over g_{3}^{2}}\int d^{3}xE_{\mu}^{a}B_{\mu}^{a}={1\over
g_{3}^{2}}\int d^{3}x\partial_{\mu}(A_{4}^{a}B_{\mu}^{a})={1\over
g_{3}^{2}}\oint\limits_{S^{2}}d^{2}s^{\mu}A_{4}^{a}B_{\mu}^{a},$ (9)

showing that $S_{BPS}$ can be written as an integral over an $S^{2}$ at
Euclidean space-time infinity. Thus, the minimum value of the action is
determined by boundary conditions at infinity, determined by the behaviour of
the Higgs field vev $v$ and “magnetic” field $B_{\mu}^{a}$ at infinity.

Let us concentrate on the simplest self-dual Euclidean solutions of finite
action, the spherically symmetric ones. We shall now give their form and in
later Sections discuss their relation to the static ’t Hooft-Polyakov monopole
solutions on $R^{1,3}$ (it is, in fact, this relation that has made the name
“monopole-instantons” stick to the instanton solutions of the 3d Polyakov
model). The solutions are known as the “BPS monopoles.” The solution centered
at the origin $r=0$ of $R^{3}$ is

$A_{4}^{a}=-n_{a}vP(vr)\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ A_{\mu}^{a}=\epsilon_{a\mu\nu}n_{\nu}{1-A(vr)\over
r}\leavevmode\nobreak\ .$ (10)

Here, $n_{a}={r^{a}\over r}$ is a unit vector in $R^{3}$, and you should
imagine that all indices are lifted and lowered with Kronecker delta’s. The
functions $P(x)=\coth x-{1\over x}\rightarrow 1-{1\over x}$,
$A(x)={x\over\sinh x}\rightarrow{\cal{O}}(xe^{-x})$, where both limits are as
$x\rightarrow\infty$. For the BPS self-dual monopole-instanton (10), the field
strength can be straightforwardly calculated. It is given in terms of two
functions $F_{1}(v,r)={v^{2}\over\sinh vr}\left({1\over vr}-\coth
vr\right)\rightarrow v^{2}{\cal{O}}(e^{-vr})$ and
$F_{2}(v,r)={v^{2}\over\sinh^{2}vr}-{1\over r^{2}}\rightarrow-{1\over r^{2}}$,
so that

$B_{\mu}^{a}=E_{\mu}^{a}=(\delta_{\mu
a}-n_{\mu}n_{a})F_{1}(v,r)+n_{\mu}n_{a}F_{2}(v,r)\rightarrow-{n_{\mu}n_{a}\over
r^{2}}\leavevmode\nobreak\ {\rm as}\leavevmode\nobreak\
r\rightarrow\infty\leavevmode\nobreak\ .$ (11)

Notice that the solution has a characteristic core size, of order $v^{-1}$,
inside which the field configuration is nonabelian and quite complicated.
However, as $r\gg v^{-1}$, the fields drastically simplify, as shown above.
The solution given above is everywhere regular and is known as the “hedgehog”
gauge solution. This is due to the fact that the isospin orientation of the
$B$ field at spatial infinity varies as a function of direction. The action of
the BPS monopole is easily computed from (10,11) and (9) as an integral over
spatial infinity

$S_{0}={1\over
g_{3}^{2}}\oint\limits_{S^{2}}d^{2}s^{\mu}A_{4}^{a}B_{\mu}^{a}={v\over
g_{3}^{2}}\oint\limits_{S^{2}}d^{2}s^{\mu}{n_{\mu}\over r^{2}}={4\pi v\over
g_{3}^{2}}\leavevmode\nobreak\ .$ (12)

Notice that the action is large in the semiclassical limit $v\gg g_{3}^{2}$.
The anti-BPS solution is given by the same equations (10, 11), but with
$A_{\mu}^{a}$ and $B_{\mu}^{a}$ taken with opposite sign.

> Exercise 1: Verify that (10) solve the equations of motion of (5) and that
> the field strength is given by (11).

Figure 2: A representation of a monopole-instanton solution in $R^{3}$,
showing the scales characterizing the $\lambda<1$ spherically-symmetric
monopole-instanton: the nonabelian core of size $v^{-1}$, the outside core
region of size $(\sqrt{\lambda}v)^{-1}$ with abelian $E_{\mu}$ field, and the
“long-range” region $r>(\sqrt{\lambda}v)^{-1}$ where only an abelian $\vec{B}$
field in the unbroken-$U(1)$ is present. For our applications, only the long-
range region will be relevant, as the monopole-instantons are well separated
in the dilute-gas approximation. The action of the $\lambda\ll 1$ solution is
approximately the BPS action (9), $S_{0}={4\pi v\over g_{3}^{2}}$, and the
magnetic charge (14) is unity.

Before we continue, let us also give the large-$r$ asymptotics of the BPS
monopole-instanton solution in a gauge where the hedgehog has been combed, the
so-called “string” gauge.202020The hedgehog can not be combed everywhere
without a singularity. One needs to cover $R^{3}$ by at least two coordinate
charts to describe the string-gauge solution for all $r,\theta,\phi$. We shall
not dwell on this as it will not be relevant for us. The formulae of the
string gauge solution covering also all $r<v$ in the two coordinate patches
can be found in e.g. [33], along with the gauge transformations relating them
to the regular hedgehog solution. In this gauge, the $B$ (and $E$) field far
away from the core of the monopole-instanton is rotated to point solely in the
3rd isospin direction, so it really resembles the field of a pointlike charge
under the unbroken $U(1)$. The $A_{4}$ and $B$ field, given now as a matrix in
the $SU(2)$ Lie algebra, approach, as $r\gg v$,

$A_{4}(r)\rightarrow{v\sigma^{3}\over 2},\leavevmode\nobreak\
\leavevmode\nobreak\ B_{r}\rightarrow{1\over r^{2}}{\sigma^{3}\over 2},$ (13)

with exponentially small angular-direction components
$B_{\theta,\phi}\rightarrow v^{2}{\cal{O}}(e^{-vr})$ (we defined the polar-
coordinate components of $B$ via
$B_{\mu}dx^{\mu}=B_{r}dr+B_{\theta}d\theta+B_{\phi}d\phi$). In this gauge, we
can write the action (9) as $S_{0}={v\over
g_{3}^{2}}\oint\limits_{S^{2}}d^{2}\vec{s}\cdot\vec{B}^{3}\equiv{4\pi v\over
g_{3}^{2}}\;Q_{m}\leavevmode\nobreak\ $, with $Q_{m}=1,$ where we defined the
“magnetic charge” in the usual way, as the integral of the “magnetic field”
$\vec{B}^{3}$ at spatial infinity

$\leavevmode\nobreak\ Q_{m}={1\over
4\pi}\oint\limits_{S^{2}}d^{2}\vec{s}\cdot\vec{B}^{3}\leavevmode\nobreak\ .$
(14)

The minimal action (anti-) BPS solutions have $Q_{m}=\pm 1$. The $Q_{m}=1$
solutions given here are also the lowest action ones. Outside of the core (of
order $v^{-1}$), the field configuration is determined by the essentially
abelian $B^{3}_{\mu}$ (and $E^{3}_{\mu}$) field. This is schematically shown
on Figure 2. We stress that the long-range abelian field, the charge, and the
action of the monopole-instanton are the main features relevant for our
discussion, while the details of the core are inessential.

Here, we presented these solutions in the BPS limit $\lambda=0$, where no
potential for the $A_{4}$ field is present. The BPS monopole-instantons also
carry long-range “electric” fields $E_{\mu}^{3}\sim n_{\mu}/r^{2}$. In the
nonsupersymmetric 3d Polyakov model and in most of our future $R^{3}\times
S^{1}$ applications, $\lambda\ll 1$ is finite, causing $A_{4}$ to become
massive. Denoting the $A_{4}$ mass by $m_{4}\sim\sqrt{\lambda}v$, the
corresponding (now non-BPS) monopole-instantons have an $E$ field that dies
exponentially away from the core, $E_{\mu}^{3}\sim
n_{\mu}{e^{-m_{4}r}/r^{2}}$, thus they carry no long-range “electric” fields.
We stress, however, the most relevant feature: the “magnetic” field remains
long-range, $B_{\mu}^{3}\sim n_{\mu}/r^{2}$, also for $\lambda\neq 0$. The
action is, as before, given by $S_{0}$ of (9), with small, for $\lambda\ll 1$,
corrections.212121When $\lambda\neq 0$, the monopole-instanton solutions are
no longer (anti-) self-dual. Exact analytic forms of the solutions do not
exist and they have to be studied numerically or by employing matched
asymptotic expansions to find approximate solutions in different regions of
space. See [33] for some expressions and references.  A cartoon, useful for
our future applications, of the field configuration at large distances is
given in Figure 2.

Summary of 2.2:

The (anti-) BPS monopole-instanton solutions (10) are finite action (9)
Euclidean solutions of the classical equations of motion of the $\lambda=0$
Polyakov model (1). Most importantly for our application, even for nonzero
$\lambda\ll 1$, these instantons carry long-range “magnetic” fields
$B_{\mu}^{3}\sim n_{\mu}/r^{2}$ and thus have “magnetic” charges (14). Their
action $S_{0}$ is given by (9), up to small-$\lambda$ corrections.

###  2.3 Monopole-instantons and the IR: I. first pass.

Armed with the knowledge of finite action instantons in the Polyakov model we
now return to study the IR physics in the semiclassical $v\gg g_{3}^{2}$
regime. Recall from Section 2.1 that, perturbatively, the IR theory is the
rather boring one of a massless photon and nothing else.

In the semiclassical limit the theory is weakly coupled and we expect, using
our intuition from quantum mechanics, that summing up the contributions of the
trivial and nontrivial saddle points of the path integral will give a good
guidance to the physics.222222To use quantum mechanics as a motivation, recall
the example of the double well potential [24, 34, 35, 25, 36]. The energy of
the ground state receives perturbative contributions calculated using an
expansion around a minimum. At any finite order, these corrections are blind
to tunneling and the existence of the second minimum. There are also
nontrivial saddle points: instantons, anti-instantons, instanton-anti-
instantons, etc. When summed over, their contributions imply that the true
ground state is the symmetric linear combination of states built around the
two minima. The energy splitting between the symmetric and anti-symmetric
combinations is also determined. There is a lot of structure here, which in
many cases can be made quite precise, see the review [22], relevant to the
fascinating and difficult topic of “resurgence” theory (that we can’t go
into).  Saddle points are classical solutions extremizing the action
functional. Ordinary perturbation theory, as in Section 2.1, is an expansion
in small fluctuations around the trivial saddle point of the path integral,
i.e. around the classical solution $A_{4}=v$, $A_{\mu}=0$ with action $S=0$.
However, in Section 2.2 we found that there are other saddle points of the
path integral, the monopole-instantons with action (9) $S_{0}={4\pi v\over
g_{3}^{2}}\gg 1$. The natural question that arises, then, is whether and how
these other saddles affect the IR physics?

We shall begin answering this question using the Euclidean path integral
framework. We found the nontrivial solutions of the equations of motion, the
monopole-instantons with action $S_{0}={4\pi v\over g_{3}^{2}}\gg 1$. Next, we
recall the equivalence between Euclidean field theory (here obtained from a
$d=2+1$ quantum field theory) and classical statistical mechanics (in $d=3$).
It implies that the Euclidean path integral can be given a probability
interpretation: any field configuration will occur with probability
proportional to its “Boltzmann” factor $e^{-S}$. Thus, we expect that in the
Euclidean path integral of the Polyakov model, such as the one representing
the vacuum to vacuum amplitude, monopole-instanton field configurations can
contribute, with probability governed by their action, $\sim e^{-S_{0}}$. A
monopole-instanton is a field configuration in the $R^{3}$ spacetime, which is
characterized by its location (the position of its center) and characteristic
core size $\sim v^{-1}$, or core volume $v^{-3}$. Such a fluctuation can
spontaneously appear anywhere in $R^{3}$. If we take a box in $R^{3}$ of
volume $V$, the classical statistical mechanics interpretation leads one to
expect that a monopole-instanton fluctuation in the path integral can appear
with probability ${Vv^{3}}e^{-S_{0}}$, where $Vv^{3}$ represents the “entropy”
enhancement and $e^{-S_{0}}$ the “energy” suppression. Thus, the probability
of a monopole-instanton fluctuation per unit spacetime volume is $\sim
v^{3}e^{-S_{0}}$.

To get an idea about the effect of this nonperturbative fluctuation, let us
study the gauge-invariant two-point field-strength correlation function, say
$\langle B_{\mu}^{3}(\vec{x})B_{\mu}^{3}(\vec{y})\rangle$. The perturbative
contribution is easily evaluated, as per Section 2.1,

$\langle
B_{\mu}^{3}(\vec{x})B_{\mu}^{3}(\vec{y})\rangle\big{|}_{pert.}\sim{g_{3}^{2}\over|\vec{x}-\vec{y}|^{3}}\leavevmode\nobreak\
.$ (15)

This is simply the contribution of the massless photon and the r.h.s. can be
guessed by dimensional analysis (once again, the coefficient is calculable but
inessential). As per our discussion above, a monopole-instanton fluctuation
can also appear and contribute to this two-point function. Suppose now that
such a fluctuation appears at some point $\vec{r}\in R^{3}$, with (per unit-
volume) probability $\sim v^{3}e^{-S_{0}}$. As in the quantum mechanics
instanton calculus, we sum over $d^{3}r$, the possible positions of the
instanton (as the fluctuation is equally likely to appear anywhere, at least
in this leading approximation). We also recall that for an instanton at
$\vec{r}$, the long-range “magnetic” field is
$\vec{B}^{3}(\vec{x})|_{1-inst.\;at\;\vec{r}}\sim{\vec{x}-\vec{r}\over|\vec{x}-\vec{r}|^{3}}$.
Thus, we find that the 1-instanton contribution to the two-point function is
given by

$\displaystyle\langle\vec{B}^{3}(x)\cdot\vec{B}^{3}(y)\rangle\big{|}_{1-instanton}$
$\displaystyle\sim$ $\displaystyle v^{3}e^{-S_{0}}\int
d^{3}r\;\vec{B}^{3}(\vec{x})|_{1-inst.\;at\;\vec{r}}\;\cdot\vec{B}^{3}(\vec{y})|_{1-inst.\;at\;\vec{r}}$
(16) $\displaystyle\sim$ $\displaystyle{v^{3}e^{-S_{0}}\int
d^{3}r}\leavevmode\nobreak\
{\vec{x}-\vec{r}\over|\vec{x}-\vec{r}|^{3}}\cdot{\vec{y}-\vec{r}\over|\vec{y}-\vec{r}|^{3}}\leavevmode\nobreak\
\sim\leavevmode\nobreak\
{v^{3}e^{-S_{0}}\over|\vec{x}-\vec{y}|}\leavevmode\nobreak\ .$

To get the last equality, one can calculate the integral over $d^{3}r$, or
simply recall from electrostatics that it represents the interaction energy of
two electric charges at $\vec{x}$ and $\vec{y}$.

Let us now add the perturbative (15) and 1-instanton (16) contribution to the
two-point function

$\langle
B_{\mu}^{3}(\vec{x})B_{\mu}^{3}(\vec{y})\rangle\big{|}_{pert.+1-inst.}\sim
g_{3}^{2}\left({1\over|\vec{x}-\vec{y}|^{3}}+Ce^{-{4\pi v\over
g_{3}^{2}}}\;{v^{2}\over|\vec{x}-\vec{y}|}\right)\leavevmode\nobreak\ ,$ (17)

where $C$ is a dimensionless constant (dependent on $g_{3}^{2}/v$) that also
incorporates a proper integration over the collective coordinates of the
monopole-instanton (this detail shall not concern us here).

Most important for what follows is the structure of (17). In the second term,
we restored the explicit small-coupling dependence of $S_{0}$. This emphasizes
the fact that the two terms have very different behaviour as
$g_{3}^{2}/v\rightarrow 0$: the second term in (17) is nonperturbative, i.e.
nonanalytic in the small dimensionless parameter. The most important
conclusion that one can draw from the two-point correlator (17) is that,
despite its exponential smallness in the ${g_{3}^{2}\over v}\ll 1$ limit, the
nonperturbative term dominates the long-distance behaviour of the two-point
function, as it becomes more important for exponentially large separations,
$|\vec{x}-\vec{y}|^{2}\gg v^{-2}e^{4\pi v\over g_{3}^{2}}$. Thus, the effect
of nonperturbative fluctuations is to significantly alter the IR physics. The
behaviour of (17) also indicates that a nonperturbative mass scale appears in
the IR theory, given by the inverse of the square root of the crossover
distance indicated above

$m_{IR}^{2}\sim v^{2}e^{-S_{0}}=v^{2}e^{-{4\pi v\over
g_{3}^{2}}}\leavevmode\nobreak\ .$ (18)

The appearance of an IR mass scale also makes one suspect that the true
behaviour of the correlator (17) at large distances may have an exponential
rather than a powerlaw falloff.

In what follows, we shall argue that the above guess is essentially correct.
We shall show that the IR physics of the Polyakov model is even “more boring”
than the free massless photon of Section 2.1—it is, instead, gapped, with a
nonperturbative mass gap given by $m_{IR}$. Perhaps some of the boredom will
disappear when we show that, upon insertion of fundamental probe electric
charges, the same mechanism that gave rise to a nonvanishing mass gap,
$m_{IR}$, is also responsible for the confinement of these charged probes.

However, before we get there, we must address the deficiencies in our
“derivation” of (17). Let’s first recall the relevant lessons we learned.
Think of the monopole-instantons as particles of size $1/v^{3}$ and a
“Boltzmann” suppression factor $e^{-S_{0}}$. In a box of 3-volume $V$, a
monopole-instanton can appear with probability ${V\over(1/v)^{3}}e^{-S_{0}}$,
indicating the “energy” suppression $e^{-S_{0}}$ and “entropy” enhancement
$Vv^{3}$. Note that for large $Vv^{3}$ the entropy contribution wins over the
Boltzmann suppression and one has to account for the appearance of multiple
monopole-instanton fluctuations. This is one of the main points we need to
address to improve upon (17). In the quantum mechanics double-well problem,
this is addressed by summing up multiple instanton contributions using the
dilute instanton gas approximation, and we shall do a similar summation here.

In the model at hand, one encounters another issue not present in quantum
mechanics, due to the long-range instanton interactions. One expects the
probability that two such objects appear to scale as
$v^{6}e^{-2S_{0}}e^{-S_{inter.}}$. Here, $S_{inter.}$ is due to the fact that
the monopole-instantons have a long-range “magnetic” field and hence non-
negligible interactions at long distances (as opposed to the double-well
instantons in quantum mechanics which have exponentially suppressed long-
distance interactions). Thus, the action of a pair of such objects taken some
distance apart will be larger or smaller than $2S_{0}$, depending on whether
they attract or repel. The long-range interactions between two monopole-
instantons is due to their long-range $B_{\mu}^{3}$ tail and we expect them to
have Coulomb-like interactions at large separations (recall the end of Section
2.2, where we argued that at $\lambda\neq 0$, the $E_{\mu}^{3}$ field is
exponentially damped away from the instanton core, as pictured on Figure 2).

Thus, in order to properly calculate correlation functions like (17), we have
to account for the contribution of multiple monopole-instantons (as in quantum
mechanics) and for their long-range interactions (a feature not present in
quantum mechanics). The picture that will emerge is that the sum over saddle
points of the Euclidean path integral of the Polyakov model can be recast in
the form of the partition function of a classical gas of monopole-instantons
and anti-monopole-instantons, with pairwise Coulomb interactions at large
separations. This classical gas is “grand-canonical,” i.e. the number of
either monopole-instantons and anti-monopole-instantons is not fixed. What is
fixed is the “fugacity” $e^{-S_{0}}$ that each object’s contribution to the
partition function is weighted by. If the gas is sufficiently dilute,
accounting for only the long-range interactions should suffice to describe the
physics, as the various monopole-instantons never significantly overlap. The
semiclassical expansion parameter controlling the diluteness of the gas is now
$e^{-S_{0}}\ll 1$.

Summary of 2.3:

In this Section, we saw the first indication that monopole-instanton
fluctuations drastically alter the perturbative IR physics of the Polyakov
model, leading to the appearance of a new infrared scale (18). However, to
proceed, we need to develop more technology to properly study the effect of
multiple instantons and their interactions.

###  2.4 Monopole-instantons and the IR: II. duality, ’t Hooft vertices, and
monopole operators.

The IR physics of the 3d Polyakov model is most conveniently described using a
dual language.232323Recently, 3d dualities more general than the one we use in
these notes have received some attention. For a pedagogical introduction and
review, see [37]. The dual description employs the fact that in 3d, a photon
has one polarization and is thus equivalent to a scalar field. The duality
that we shall perform is thus valid in the IR theory (3) of the UV $SU(2)$
model (1). Recall that to obtain the IR theory, the $W$-bosons are integrated
out, and the only light degree of freedom is $A_{\mu}^{3}$ (and possibly the
$A_{4}^{3}$ neutral scalar, which remains massless for $\lambda=0$). The long
distance theory is defined by the action (3),

$L_{eff.}={1\over 4g_{3}^{2}}F_{\mu\nu}^{2},$ (19)

and a path integral over the $A_{\mu}$ field modulo gauge transformations. For
brevity, we shall henceforth omit the isospin index on the fields $A_{\mu}$,
$F_{\mu\nu}$, with the understanding that these are the massless modes
describing the unbroken $U(1)$ subgroup of $SU(2)$. There are no charged
degrees of freedom in this theory and the path integral over $A_{\mu}$ can be
rewritten as a path integral over $F_{\mu\nu}$, with an additional constraint
imposed to ensure that the field strength obeys the Bianchi identity
$\epsilon^{\mu\nu\lambda}\partial_{\mu}F_{\nu\lambda}=0$ (which guarantees
that locally $F$ is the curl of a vector). The constraint can be imposed via a
Lagrange multiplier scalar field $\sigma$. To avoid the appearance of factors
of $i$ in the action, we shall now go to Minkowski space with metric $(+,-,-)$
and $\epsilon^{012}=+1$. The action is

$S_{Mink.}[F_{\mu\nu},\sigma]=-{1\over 4g_{3}^{2}}\int
d^{3}xF_{\mu\nu}F^{\mu\nu}-{1\over 8\pi}\int
d^{3}x\;\partial_{\mu}\sigma\;F_{\nu\lambda}\epsilon^{\mu\nu\lambda}$ (20)

and the path integral is over $F_{\mu\nu}$ and $\sigma$. Integrating out
$\sigma$ imposes the Bianchi identity for $F_{\mu\nu}$, i.e. gives back the
Minkowski space version of the original theory (19). On the other hand,
integrating out $F_{\mu\nu}$, and substituting back into (20) its saddle point
value

$F^{\mu\nu}=-{g_{3}^{2}\over
4\pi}\epsilon^{\mu\nu\lambda}\partial_{\lambda}\sigma,$ (21)

we obtain the “dual photon” action

$S_{Mink.}[\sigma]={1\over 2}{g_{3}^{2}\over(4\pi)^{2}}\int
d^{3}x\partial_{\lambda}\sigma\partial^{\lambda}\sigma\leavevmode\nobreak\ .$
(22)

The field $\sigma$ will be called, from now on, the “dual photon.” The
description in terms of the dual photon is often called the “magnetic
description,” in contrast with the electric description in terms of a $U(1)$
gauge field.

The duality relation (21) shows how to map local gauge invariant operators
between the electric description (19) and the magnetic one (22). For future
use, we note that the spatial gradients of $\sigma$ give the $F^{0i}$,
$i=1,2$, components of the field strength, i.e. they represent the true
electric field on the $R^{1,2}$ Minkowski space. The time derivative of
$\sigma$, on the other hand, represents the $F^{12}$ component, i.e. the only
magnetic field component on $R^{1,2}$.

There are, however, other operators that will be of interest to us, that may
be slightly less familiar. Begin, in the electric theory (19), where one can
define line operators representing the insertion of static nondynamical charge
probes, such as

$W_{Q_{e}}(\vec{x}_{*})=e^{iQ_{e}\int dx^{0}A_{0}(\vec{x}_{*},x^{0})},$ (23)

the Wilson line representing a static electric charge $Q_{e}$ located at
$\vec{x}_{*}\in R^{2}$. Such operators are meant to be inserted in the path
integral and are important to the study of confinement. Thus, it will be of
interest to us to learn how Wilson lines are represented in the dual theory
(22).

Conversely, in the magnetic theory (22), one can define operators that create
“fluxons.” To justify their introduction, notice that the dual photon theory
(22) has a global symmetry under which $\sigma$ shifts by a constant. The
corresponding current is $j_{\mu}={g_{3}^{2}\over
16\pi^{2}}\partial_{\mu}\sigma$, which is mapped, by (21) to $j_{\mu}=-{1\over
8\pi}\epsilon_{\mu\nu\rho}F^{\nu\rho}$. Thus, the shift symmetry of the dual
photon maps to the “0-form” magnetic symmetry of the electric 3d $U(1)$
theory, whose current is conserved due to the Bianchi identity. We note that
this magnetic symmetry is an “emergent” symmetry of the IR $U(1)$ theory, and
is not present in the $SU(2)$ UV theory of the Polyakov model. In fact, its
breaking by the UV $SU(2)$ theory is crucial for the physics of
confinement.242424This is also not a symmetry in the compact-$U(1)$ lattice
theory which also exhibits confinement [38, 25]. For a study of the compact
$U(1)$ theory, including in the Hamiltonian formalism, see [39].

The charge corresponding to the $\sigma$-shift symmetry is

$Q={g_{3}^{2}\over 16\pi^{2}}\int
d^{2}x\partial_{0}\sigma\leftrightarrow-{1\over 4\pi}\int d^{2}xF_{12},$ (24)

and maps to the integral over $R^{2}$ of the magnetic field $F_{12}$. In a
canonical quantization of the $\sigma$ theory (22), the state

$e^{i\sigma(\vec{x}_{*})}|0\rangle$ (25)

is an eigenstate of $Q$ with eigenvalue $1$, as you will show in the following

> Exercise 2: Canonically quantize the dual theory (22) (for brevity, omitting
> hats over operators). Show that the canonical commutation relations imply
> that $Qe^{\pm i\sigma(\vec{x}_{*})}|0\rangle=\pm e^{\pm
> i\sigma(\vec{x}_{*})}|0\rangle$, i.e. the operator $e^{\pm
> i\sigma(\vec{x}_{*})}$ creates a unit magnetic flux, a “fluxon” at
> $\vec{x}_{*}\in R^{2}$.

Notice that in the electric description (19), the operator $e^{i\sigma}$,
creating a pointlike fluxon with $-{1\over 4\pi}\int d^{2}xF_{12}=1$ does not
have a simple description. This is an example of a “disorder” operator which
does not have a local expression in terms of the electric theory fields.

Figure 3:  The photon-dual photon duality (21) maps an electric charge $Q_{e}$
into a vortex of the dual photon field, with monodromy determined by the
charge. As shown in Exercise 3, the monodromy equals $\pm 2\pi$ for “quarks”
in the fundamental of $SU(2)$. A fundamental Wilson loop is thus mapped to a
disorder operator for the $\sigma$ field, defined by imposing $2\pi$ monodromy
of the dual photon around the loop.

Conversely, the Wilson loop operator (23) is simple in the electric theory,
but not in the magnetic theory (22), where it is represented by a disorder
operator. As Wilson loops are important for the study of confinement, let us
now flesh out the details. As already discussed $W_{Q_{e}}(\vec{x}_{*})$
describes the insertion at $\vec{x}_{*}$ of a static electric probe charge
$Q_{e}$. A static charge at the origin of $R^{2}$ creates electric field
$\vec{\cal{E}}(\vec{r})\sim Q_{e}{\vec{r}\over r^{2}}$ (for the
proportionality constant, consult Exercise 3 below). By Gauss’ law, a line
integral of the electric field over a loop $C\in R^{2}$ enclosing the charge
can be used to find the charge, $Q_{e}\sim\oint\limits_{C}\vec{\cal{E}}\cdot
d\vec{s}$, with $\vec{s}$ normal to $C$. From the duality relation (21), we
have that $Q_{e}\sim\oint\limits_{C}\vec{\cal{E}}\cdot
d\vec{s}\sim\oint\limits_{C}d\vec{l}\cdot\vec{\nabla}\sigma$, where $\vec{l}$
is tangent to $C$. In other words, we have shown that the $\sigma$ field has
nonvanishing monodromy,252525A nonzero monodromy simply means that $\sigma$ is
not single-valued around electric charges: taking $C$ to be a circle
parametrized by a polar angle, we have
$\oint_{C}d\vec{l}\cdot\vec{\nabla}\sigma=\sigma(2\pi)-\sigma(0)\sim Q_{e}$.
proportional to $Q_{e}$, around loops surrounding static electric charges, see
Figure 3. Thus, the Wilson loop operator $W_{Q_{e}}(\vec{x}_{*})$ should be
defined by the following prescription: when inserted in the path integral of
the dual photon theory, impose boundary conditions on the $\sigma$ field in
the path integral, requiring it to have monodromy $\sim Q_{e}$ along the line
$\vec{x}=\vec{x}_{*}$ in $R^{3}$. To determine the coefficient, consider the
following

> Exercise 3.1: Our electric theory (19), which descends from an $SU(2)$ gauge
> theory (1), is an example of a compact $U(1)$ theory—electric charge under
> the unbroken $U(1)$ is quantized, with the smallest value corresponding to
> sources in the fundamental representation of $SU(2)$.262626To do this
> problem, we use the fact that a static charge in the fundamental
> representation of $SU(2)$ can be introduced by inserting a fundamental
> Wilson line, $W(\vec{x}_{*})=\text{tr}\;{\cal{P}}e^{i\int
> dx^{0}A_{0}^{a}(\vec{x}_{*},x^{0})T^{a}}$ in the path integral, see the
> discussion around (43). This leads, in the unbroken-$U(1)$ theory, to the
> insertion of the sum of two operators like (23). Here, ${\cal{P}}$ denotes
> path ordering, which can be ignored when projecting (ignoring the massive
> components of $A_{\mu}$) to the $U(1)$ sector. Thus, introducing a static
> fundamental charge corresponds to adding a term $\int
> dx^{0}A_{0}^{3}(\vec{x}_{*},x^{0})(T^{3})_{ii}$, with $i=1$ or $i=2$, to the
> Minkowski space version of the IR theory action (19). Here $(T^{3})_{ij}$
> denotes the $ij$-th entry of the Cartan generator of $SU(2)$.  Notice that
> the insertion of a static charge in the fundamental representation of
> $SU(2)$ at $\vec{x}_{*}\in R^{2}$ corresponds to adding $\int
> dx^{0}A_{0}^{3}(\vec{x}_{*},x^{0})(T^{3})_{ii}$, with $i=1$ or $i=2$, to the
> Minkowski-space unbroken-$U(1)$ theory action. Following the above
> discussion, be mindful of the coefficients and show that the monodromy of
> the $\sigma$ field around a static charge in the fundamental of $SU(2)$ is
> $\pm 2\pi$.
>
> Exercise 3.2: The result of this Exercise can be used to further the study
> of the deconfinement transition briefly described in Section 4.3. We include
> it here because it may help solidify the understanding of the charge-vortex
> duality of Figure 3. Consider the following static $\sigma$-field
> configuration:
>
>
> $\sigma(\vec{r})=q_{1}\theta(\vec{r}-\vec{r}_{1})+q_{2}\theta(\vec{r}-\vec{r_{2}})\leavevmode\nobreak\
> ,$ (26)
>
> where $\theta(\vec{r})$ is the angle the vector $\vec{r}\in R^{2}$ makes
> with, say, the positive-$x$ axis. This “two-vortex” configuration has
> $\sigma$ monodromy $2\pi q_{1}$ around $\vec{r}_{1}$ and $2\pi q_{2}$ around
> $\vec{r}_{2}$. Thus, according to the duality, it represents two electric
> charges. As per Exercise 3.1, if $|q_{1,2}|=1$, these are fundamental
> charges. While the angle $\theta(\vec{r})$ is not defined at $\vec{r}=0$,
> showing that a UV definition is needed (like a lattice cutoff), this does
> not affect the calculation of the interaction energy between the charges.
> Use the dual photon action (22) to calculate the interaction energy between
> the charges represented by (26). Letting $\vec{R}=\vec{r}_{1}-\vec{r}_{2}$,
> show that the interaction energy is
>
> $E(R)=-q_{1}q_{2}\;{g_{3}^{2}\over 8\pi}\;\log{R\over a}\leavevmode\nobreak\
> ,$ (27)
>
> such that e.g. like charges logarithmically repel (here $a$ is some
> irrelevant short distance cutoff needed to define the vortex
> configurations). As a sanity check, also convince yourself that for
> $|q_{1}|=|q_{2}|=1$, the same interaction energy between static fundamental
> charges can be obtained from the electric theory. (There are various ways to
> do this problem, but the relation
> $\partial_{i}\theta(\vec{r})=\epsilon_{ij}\partial_{j}\log|\vec{r}|$ and
> judicial integration by parts helps.)

The conclusion from Exercise 3 is that, to permit the insertion of static
fundamental charge probes, which are natural in the UV $SU(2)$ theory, $2\pi$
monodromies of the $\sigma$ field should be allowed. These only make sense if
the dual photon field is regarded as a compact scalar field,
$\sigma\equiv\sigma+2\pi$, i.e. the field $\sigma$ is a map from the $R^{3}$
spacetime to the $S^{1}$ field space. We also notice that the $2\pi$
periodicity of $\sigma$ is consistent with the $e^{i\sigma}$ being a well-
defined operator (while a $\pi$ periodicity would not be consistent).

In Exercise 2, you showed that in the canonical picture the operator
$e^{i\sigma}$ creates “fluxons,” pointlike excitations for which $-{1\over
4\pi}\int d^{2}xF_{12}$ $=$ $1$. Notice that the unit magnetic flux of the
“fluxon” just given is equal to the magnetic charge of the monopole-instanton
(14) with $|Q_{m}|=1$ (recall that $B_{3}^{3}=F_{12}$). This suggests that
magnetic monopole-instantons of unit charge (14) can be interpreted as
tunneling events between states differing by one flux quantum. This can be
argued by deforming the integral over $S^{2}$ in (14) to equal the difference
between two $R^{2}$-integrals like (24), specifying the number of fluxons in
the states at the initial and final time $\pm T$ (as stressed in [28]). That
the number of fluxons can change reflects the fact that (24) is not conserved
in the full theory.

In fact, we shall now argue an important relation valid for the Polyakov model
and also useful for our future $R^{3}\times S^{1}$ studies: in the Euclidean
path integral of the dual-photon theory, insertions of $e^{\pm i\sigma(x)}$,
with $x\in R^{3}$, represent the appearance, at $x$, of a monopole-instanton
of minimal “magnetic” charge (14), discussed in Section 2.2.272727To avoid
confusion, the object $e^{\pm i\sigma}$ represents the insertion of a
pointlike object, i.e. the structure of the monopole-instanton inside its core
($\sim v^{-1}$) is ignored. This suffices in the dilute gas approximation
where only the long-distance monopole-instanton interactions are important. To
begin, define the following object (usually called generating functional)

$\langle e^{i\int
d^{3}x\rho(x)\sigma(x)}\rangle\equiv\zeta^{-1}\int{\cal{D}}\sigma\;e^{-{\kappa\over
2}\int d^{3}x(\partial_{\mu}\sigma)^{2}+i\int
d^{3}x\rho(x)\sigma(x)}\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \text{where}\;\kappa\equiv{g_{3}^{2}\over(4\pi)^{2}},$
(28)

and $\zeta$ is a normalization factor ensuring that the l.h.s. equals unity
when $\rho=0$. It is a standard result that $\langle e^{i\int
d^{3}x\rho(x)\sigma(x)}\rangle=e^{-{1\over 8\pi\kappa}\int
d^{3}xd^{3}x^{\prime}\rho(x){1\over|x-x^{\prime}|}\rho(x^{\prime})}$. We next
concentrate on a particular form of $\rho(x)=\rho_{N}(x)$:

$\rho_{N}(x)=\sum_{a=1}^{N}q_{a}\delta^{(3)}(x-x_{a}),$ (29)

where $q_{a}$ are integers. Notice that $\rho_{N}(x)$ can be interpreted as
the charge density at $x$ of a gas of $N$ pointlike charges $q_{a}$ located at
$x_{a}\in R^{3}$.

> Exercise 4: Show that,282828For future use, note that integration over the
> zero mode of $\sigma$ gives
> $\int\limits_{0}^{2\pi}d\sigma_{0}e^{i\sigma_{0}\sum\limits_{a=1}^{N}q_{a}}$.
> For integer $q_{a}$ this is only nonzero provided
> $\sum\limits_{a=1}^{N}q_{a}=0$. with $\rho(x)=\rho_{N}(x)$,
>
> $\langle e^{i\int d^{3}x\rho(x)\sigma(x)}\rangle=e^{-{1\over
> 4\pi\kappa}\sum\limits_{a>b}{q_{a}q_{b}\over|x_{a}-x_{b}|}}\leavevmode\nobreak\
> ,$ (30)
>
> where divergent terms with $a=b$ have been omitted (these divergences are to
> be absorbed in the UV definition of the monopole-instanton fugacities, see
> further below). As a corollary, show that with the same omission,
>
> $\langle e^{iq_{1}\sigma(x_{1})+iq_{2}\sigma(x_{2})}\rangle=e^{-{4\pi\over
> g_{3}^{2}}{q_{1}q_{2}\over|x_{1}-x_{2}|}}.$ (31)

To interpret the last relation, we now do the following

> Exercise 5: Consider two monopole-instantons of Section 2.2, one at
> $x_{1}\in R^{3}$ and the other at $x_{2}\in R^{3}$. According to the
> discussion there, for $\lambda\ll 1$ we ignore their $E_{\mu}^{3}$
> fields.292929These extra interactions present for $\lambda=0$ have to be
> accounted for in SYM. We shall do so when we discuss the supersymmetric
> case. Notice that the calculation of the “electric” long-range interactions
> present when $\lambda=0$ is significantly more involved than Exercise 5 (see
> the discussion in Section 2.3 of [40] and references therein). However, the
> result can be stated simply. Let the monopole-instantons have magnetic
> charges $q_{i}=\pm 1$, $i=1,2$. Let also $|x_{1}-x_{2}|\gg v^{-1}$, so that
> the cores of the monopole-instantons do not overlap. Outside their cores the
> field reduces to the sum of the abelian magnetic monopole terms,
> $B_{\mu}^{3}(y)\simeq q_{1}B_{\mu}^{3}(y-x_{1})+q_{2}B_{\mu}^{3}(y-x_{2})$
> (with $B_{r}^{3}=1/r^{2}$ around each monopole-instanton). Compute the
> interaction action of the two monopole-instantons by calculating the
> contribution to the action $S={1\over 2g_{3}^{2}}\int
> d^{3}y(B_{\mu}^{3}(y))^{2}$ from the region outside the cores, extract the
> interaction term, and show that
>
> $e^{-S_{inter.}}=e^{-{4\pi\over g_{3}^{2}}{q_{1}q_{2}\over|x_{1}-x_{2}|}},$
> (32)
>
> exactly reproducing the r.h.s. of (31).
>
> Notice that the above result has a simple intuitive explanation: two
> magnetic monopoles, considered as static particles in $R^{3}$, interact via
> a magnetic version of the Coulomb law. If $q_{1}$ and $q_{2}$ are of the
> same sign, the probability to find the two charges close to each other,
> controlled by $e^{-S_{inter.}}$, is vanishingly small, corresponding to
> repulsion. Conversely, if the charges have opposite signs, the probability
> grows with decreasing separation, showing that the charges attract.

The conclusion we draw from comparing (31) with (32) is that, in the long-
distance abelian IR theory, in the dilute gas regime where monopole-instantons
are sufficiently far away so that their cores do not overlap, the appearance
of a monopole-instanton of magnetic charge $q$, at $x\in R^{3}$, can be
represented by inserting $v^{3}e^{-S_{0}}e^{iq\sigma(x)}$ in the path integral
of the dual-photon theory. The $v^{3}e^{-S_{0}}$ fugacity factor was already
discussed in Section 2.3. The novelty here is that the $e^{iq\sigma(x)}$
insertions correctly account for the long-distance interactions of the
monopole-instantons of magnetic charge $q\;(=\pm 1)$.

We can now introduce the following mnemonic accounting for nonperturbative
fluctuations in the Euclidean path integral of the dual photon theory (again,
we leave SYM for later). The semiclassical saddles of lowest action are
(approximately-) BPS and anti-BPS monopole instantons of minimal charge $\pm
1$. We denote them by $M$ and $M^{*}$, respectively, and associate with them
the following “’t Hooft vertices” [41], or “monopole operators:”

$\displaystyle M\;\text{at}\;x$ $\displaystyle:$ $\displaystyle
v^{3}e^{-S_{0}}e^{i\sigma(x)}\leavevmode\nobreak\ ,$ (33) $\displaystyle
M^{*}\;\text{at}\;x$ $\displaystyle:$ $\displaystyle
v^{3}e^{-S_{0}}e^{-i\sigma(x)}\leavevmode\nobreak\ .$

In the classical 3d statistical mechanics picture, the $v^{3}e^{-S_{0}}$
factor can be interpreted as the fugacity of $M$ or $M^{*}$.303030 Similar
vertices can also be written in the electroweak sector of the 4d standard
model, where the instantons generate exponentially suppressed $B+L$ violating
interactions, as first done by ’t Hooft [41, 42]. The main difference with the
present setup is that the 4d instantons have no Coulomb-like long-range
interactions, so no analogue of the $e^{\pm i\sigma}$ factors are present.
(Nonetheless, one can account to the interactions between instantons due to
gauge field exchange by modifying the ’t Hooft vertex to include the
interactions of instantons with gauge fields, see the description in [27].)

A slight subtlety that we shall ignore is that there is additional dependence
on the dimensionless coupling constant $g_{3}^{2}/v$, which multiplies both
$M$ and $M^{*}$ above. These factors arise upon taking into account the
integration over collective coordinates and the determinants in the monopole-
instanton backgrounds and give a power-law dependence on the dimensionless
coupling of the pre-exponential factor in the ’t Hooft vertices. As this
power-law dependence can not compete (at small $g_{3}^{2}/v$) with the
exponential $e^{-S_{0}}=e^{-{4\pi v\over g_{3}^{2}}}$, we shall ignore it. We
shall often refer to the neglect of these pre-exponential terms in (33) as
working with “exponential-only accuracy.”313131For the interested reader, the
computation of these pre-exponential terms has been performed (to one-loop
order) in the greatest detail in SYM, see [21, 43]. Accounting for collective
coordinate integrations is discussed in many textbooks, e.g. [34, 35, 36, 27].

Summary of 2.4:

In this Section, we introduced a duality transformation in the abelian IR
theory of the Polyakov model, leading to the dual-photon description. We
discussed the mapping of operators between the electric and magnetic
description. We showed that the insertion of a fundamental Wilson loop
corresponds to the requirement of a $2\pi$ monodromy of the dual photon field
around the loop. We also showed that the dual photon picture allows one to
incorporate the effect of monopole-instantons in the dilute gas approximation,
accounting for their long-range interactions. In the following Sections, we
shall sum over all possible insertions of “monopole operator” (33) in the
partition function of the dual theory to find the effect of the monopole-
instanton fluctuations in the dilute gas approximation and to study the
physics of confinement.

###  2.5 Monopole-instantons and the IR: III. dilute gas and mass gap.

Now that we have done all preparatory work, we are ready to start enjoying the
fruits of our labour. Our IR theory is, at the perturbative level, defined via
a path integral of the dual photon theory with action (22). We argued in
Section 2.3 that including nonperturbative monopole-instanton fluctuations
changes the IR physics. We shall now include the fluctuations of arbitrary
numbers of monopole-instantons and anti-monopole-instantons.

Consider the probability of a fluctuation of $n$ monopole-instantons $M$ and
$m$ anti-monopole-instantons $M^{*}$ in the Euclidean path integral. Per our
discussion above (33), we expect that such a fluctuation corresponds to the
insertion in the $\sigma$-theory Euclidean path integral of the following
object

$P_{n,m}={\left(v^{3}e^{-S_{0}}\right)^{n}\over
n!}{\left(v^{3}e^{-S_{0}}\right)^{m}\over m!}\int
d^{3(n+m)}r\;e^{i\sum\limits_{a=1}^{n+m}q_{a}\sigma(r_{a})}\leavevmode\nobreak\
={\left(v^{3}e^{-S_{0}}\right)^{n}\over
n!}{\left(v^{3}e^{-S_{0}}\right)^{m}\over m!}\int d^{3(n+m)}r\;e^{i\int
d^{3}x\sigma(x)\rho_{n+m}(x)}\leavevmode\nobreak\ ,$ (34)

where $\rho_{n+m}(x)$ is defined in (29) via the $n+m$ $q_{a}$ and $r_{a}$.

As the notation is somewhat condensed, let us elaborate. Here the index $a$
runs from $1$ to $n+m$, i.e. runs over both $M$ and $M^{*}$. $q_{a}$ denote
the corresponding charges, which take values $+1$ or $-1$ (we only take into
account $q=\pm 1$, as all other monopole-instantons of higher charges have
higher action and their effect will be exponentially suppressed compared to
$e^{-S_{0}}$). Likewise, $r_{a}\in R^{3}$ are the corresponding positions of
$M$ or $M^{*}$. The measure $d^{3(n+m)}r$ should be understood to mean the
product over all $n+m$ $dr_{a}$’s. The $v^{3}e^{-S_{0}}$ factors are the
fugacities of these objects, while the term
$e^{i\sum\limits_{a=1}^{n+m}q_{a}\sigma(r_{a})}$ (written also in terms of the
charge density $\rho_{n+m}$ in the second equality above) accounts for the
long-distance Coulomb interactions of the $n+m$ $M$ and $M^{*}$, as discussed
in the previous Section. The factors of $n!$ and $m!$ take into account the
fact that these fluctuations are indistinguishable: a fluctuation of a
monopole-instanton $M$ at $r_{1}$ and another one, $M^{\prime}$ at $r_{2}$ is
the same as $M^{\prime}$ at $r_{1}$ and $M$ at $r_{2}$.

We now insert $P_{n,m}$ into the Euclidean path integral of $\sigma$, to
obtain the partition function of the IR theory accounting for the appearance
of the $n,m$ fluctuation:

$\displaystyle Z_{n,m}=$ $\displaystyle=\zeta^{-1}$
$\displaystyle\int{\cal{D}}\sigma\;e^{-{\kappa\over 2}\int
d^{3}x(\partial_{\mu}\sigma)^{2}}P_{n,m}=\zeta^{-1}{\left(v^{3}e^{-S_{0}}\right)^{n}\over
n!}{\left(v^{3}e^{-S_{0}}\right)^{m}\over m!}\int
d^{3(n+m)}r\int{\cal{D}}\sigma\;e^{-{\kappa\over 2}\int
d^{3}x(\partial_{\mu}\sigma)^{2}}\;e^{i\sum\limits_{a=1}^{n+m}q_{a}\sigma(r_{a})}$
$\displaystyle=\zeta^{-1}$ $\displaystyle\int{\cal{D}}\sigma\;e^{-{\kappa\over
2}\int d^{3}x(\partial_{\mu}\sigma)^{2}}{\left(\int
d^{3}rv^{3}e^{-S_{0}}e^{i\sigma(r)}\right)^{n}\over n!}{\left(\int
d^{3}rv^{3}e^{-S_{0}}e^{-i\sigma(r)}\right)^{m}\over m!}$ (35)

The second line is simply a rearrangement of the first, taking into account
that all $n$ $M$ terms have identical form, as do all $m$ $M^{*}$ terms.
Notice that if we integrate over $\sigma$, we find the $n+m$-particle
contribution to the grand-canonical partition function of a classical
nonrelativistic $3d$ Coulomb gas. (In fact, the treatment that we give it here
corresponds to the Debye-Hueckel approximation in the theory of charged gases
[24, 25].)

Next we want to sum over all values of $n$, $m$, i.e. over all possible $M$
and $M^{*}$ fluctuations. Recall now that in (28) we introduced the
$\zeta^{-1}$ factor simply to make the normalization convenient and that, with
$\zeta^{-1}$ present, the IR theory partition function (2.5) with $n=m=0$
would be trivial. Thus, to obtain our IR theory partition function, when
summing over $n$ and $m$ we drop this factor. The partition function of our IR
theory with all monopole-instantons summed over becomes

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\sum\limits_{n,m=0}^{\infty}\zeta
Z_{n,m}=\int{\cal{D}}\sigma\;e^{-{\kappa\over 2}\int
d^{3}x(\partial_{\mu}\sigma)^{2}}\left(\sum\limits_{n=0}^{\infty}{\left(\int
d^{3}rv^{3}e^{-S_{0}}e^{i\sigma(r)}\right)^{n}\over
n!}\right)\left(\sum\limits_{m=0}^{\infty}{\left(\int
d^{3}rv^{3}e^{-S_{0}}e^{-i\sigma(r)}\right)^{m}\over m!}\right)$ (36)
$\displaystyle=$ $\displaystyle\int{\cal{D}}\sigma\;e^{-\int
d^{3}x{\kappa\over 2}(\partial_{\mu}\sigma)^{2}}\;e^{\int
d^{3}xv^{3}e^{-S_{0}}e^{i\sigma(x)}}\;e^{\int
d^{3}xv^{3}e^{-S_{0}}e^{-i\sigma(x)}}\leavevmode\nobreak\ ,$

where, on the last line we performed the sums over $n$ and $m$.

Combining everything, we find that the IR theory partition function,
accounting for arbitrary (anti-) monopole-instantons in the dilute gas
approximation, is323232For future use, let us note an important implication of
the above summation of the dilute instanton gas contributions, which is quite
generally valid. It shows that semiclassical objects of positive fugacity (as
our monopole instantons) contribute a negative term to the ground state
energy: evaluated at $\sigma=0$, the minimum of the potential in (37), their
effect is to give a negative contribution to the ground state energy (this is
well known in quantum mechanics of the double-well potential, where tunneling
lowers the ground state energy). In the future, we shall call objects of
positive fugacity “real saddles” (see Sections 6.1 and 6.3).

$\displaystyle Z$ $\displaystyle=$ $\displaystyle\int{\cal{D}}\sigma\;e^{-\int
d^{3}x\left[{\kappa\over
2}(\partial_{\mu}\sigma)^{2}-2v^{3}e^{-S_{0}}\cos\sigma\right]}\leavevmode\nobreak\
,$ (37)

hence, the Euclidean action of the IR dual-photon theory is, after shifting
the potential by an inessential constant:333333A remark we elaborate on in
Sections 4 and 4.2.3 is due here as well. The effective action given below
contains only the leading terms in a combined perturbative expansion in powers
of $g_{3}^{2}/v$ and a semiclassical expansion in powers $e^{-S_{0}}$. The
higher-order terms in the semiclassical expansion in the Polyakov model have
not received much scrutiny on their own, but we expect that they share many
features with the ones briefly discussed for $R^{3}\times S^{1}$ theories in
Section 6.3. (N.B. As these notes were being finalized, ref. [44] appeared,
discussing these issues.)

$S_{IR}=\int d^{3}x{g_{3}^{2}\over(4\pi)^{2}}\left[{1\over
2}\partial_{\mu}\sigma\partial^{\mu}\sigma+{2v^{3}\over\kappa}e^{-{4\pi v\over
g_{3}^{2}}}(1-\cos\sigma)\right]\leavevmode\nobreak\ ,\leavevmode\nobreak\
\kappa={g_{3}^{2}\over(4\pi)^{2}}.$ (38)

Notice that the potential term generated by summing over $M$ and $M^{*}$
simply looks like adding to the dual photon action the ’t Hooft vertices of
$M$ and $M^{*}$ (33), a result due to the dilute-gas approximation.

We can now take stock and discuss what the result (38) represents. We
immediately notice that the dual photon field acquires a mass. As written, the
potential in (38) has a unique minimum at $\sigma=0$ (recall that $\sigma$ has
a $2\pi$ periodicity). Expanding the potential around the minimum, we see that
the mass of the dual photon is

$m_{\sigma}^{2}={2v^{3}\over\kappa}e^{-{4\pi v\over g_{3}^{2}}},$ (39)

in agreement with our earlier guess (18), and with exponential-only accuracy
(in fact, a more precise calculation gives an extra factor of
$v^{2}/g_{3}^{4}$ due to monopole-instanton collective coordinate
integrations; in what follows, we simply absorb these into the definition of
$m_{\sigma}$). Thus, at weak coupling, we find that an exponential scale
hierarchy is generated:

$v\gg g_{3}^{2}\gg m_{\sigma}\sim ve^{-4\pi v/g_{3}^{2}}.$ (40)

It is the separation of the various scales due to the $v\gg g_{3}^{2}$ weak-
coupling condition (recall Section 2.1) which makes our effective field theory
treatment useful. In contrast with instanton calculations in 4d QCD [36], the
weak-coupling (due to Higgsing) leads to the important difference that the
instanton size is fixed and the problem of large-size instantons does not
arise.

Let us also discuss the scales characterizing the dilute monopole-instanton
gas. The probability (per unit volume) of a monopole-instanton fluctuation
implies that the number of instantons per unit volume is $v^{3}e^{-S_{0}}$.
Thus, the typical separation between instantons is $\bar{\ell}\sim
v^{-1}e^{S_{0}/3}$. Notice that this is exponentially larger than the
$W$-boson Compton wavelength $v^{-1}$. The inverse dual-photon mass defines
another length scale $m_{\sigma}^{-1}\sim\ell_{D}$, called the Debye screening
length (this terminology is borrowed from the charged nonrelativistic plasma).
From (40), we have that $\ell_{D}\sim v^{-1}e^{S_{0}/2}\gg\bar{\ell}$, i.e.
the Debye length is exponentially larger than the typical distance
$\bar{\ell}$ between $M$ and $M^{*}$. Thus, the validity of the dilute gas
treatment can be rephrased by requiring that there be a large number of $M$
and $M^{*}$ in a Debye volume.

> Exercise 6:
>
> Further properties of the system can be studied, using, instead of (37), the
> following
>
> $\displaystyle Z[\eta]$ $\displaystyle\equiv$
> $\displaystyle\int{\cal{D}}\sigma\;e^{-\int d^{3}x\left[{\kappa\over
> 2}(\partial_{\mu}(\sigma-\eta))^{2}+m_{\sigma}^{2}(1-\cos\sigma)\right]}\leavevmode\nobreak\
> .$ (41)
>
> Reverse the arguments leading to (37) to show that a nonzero $\eta$
> corresponds to the insertion of $e^{i\int d^{3}x\eta(x)\rho_{n+m}(x)}$ in
> the sum over monopole instantons. In other words, argue that $Z[\eta]$ is a
> generating functional for correlation functions of the monopole-instanton
> magnetic charge density. Show that the average charge density vanishes,
> $\langle\rho(x)\rangle=-i{\delta
> Z[\eta]\over\delta\eta(x)}\big{|}_{\eta=0}=0$ and find the scale of
> exponential fall-off of the two-point function
> $\langle\rho(x)\rho(y)\rangle$.

Figure 4:  The hierarchy, controlled by the exponentially large $e^{S_{0}}$,
between the various length scales characterizing the the dilute $M$-$M^{*}$
monopole-instanton gas. The dual photon Compton wavelength (the Debye
screening length) is the largest length scale. A Debye volume contains a large
number of $M$ and $M^{*}$ fluctuations.

A final comment, before we continue to discuss confinement, relates to
terminology. One often hears that confinement is due to “magnetic monopole
condensation,” in a manner dual to the usual Higgs mechanism. While this is
demonstratively true in the Seiberg-Witten theory, where magnetic monopoles
are particle-like physical excitations, in the Polyakov model, as well as in
its $R^{3}\times S^{1}$ generalizations, this terminology should be taken with
a grain of salt. The “monopoles” discussed here, leading to the mass gap
generation, are instantons. They are pseudoparticles (in ’t Hooft’s
terminology [41]) localized in spacetime, thus existing only “for an instant.”
It is the proliferation of these Euclidean monopole-instanton fluctuations in
the vacuum that leads to the mass gap and, as we show next, to
confinement.343434The relation of the monopole-instantons on $R^{3}\times
S^{1}$ to physical monopole particles on $R^{4}$ whose condensation (in
Seiberg-Witten theory) leads to a dual superconductivity is, at best, far from
straightforward, see the discussion in [18].

To end this Section, we rewrite (38) in terms of the dual photon mass
$m_{\sigma}$, absorbing the inessential (for the present discussion)
coefficient into its definition, a form that we shall use from now on:

$S_{IR}=\int d^{3}x{g_{3}^{2}\over(4\pi)^{2}}\left[{1\over
2}\partial_{\mu}\sigma\partial^{\mu}\sigma+m_{\sigma}^{2}(1-\cos\sigma)\right]\leavevmode\nobreak\
.$ (42)

Summary of 2.5:

The major result obtained here is the IR effective action for the dual photon
field (42). It shows that calculable nonperturbative effects at weak coupling
generate an exponentially small mass gap $m_{\sigma}$ (39). The exponential
hierarchy of scales (40) arises due to $v\gg g_{3}^{2}$ and is responsible for
calculability and for the utility of the effective field theory approach to
describing the long-distance physics.

###  2.6 Monopole-instantons and the IR: IV. confinement and the string
tension.

We have come close to the end of our discussion of the Polyakov model. To
proceed, as indicated in the Introduction, we first need to define what we
mean by “confinement.” The most precise definition of confinement applies to
theories without fundamental-representation dynamical fields, such as pure
Yang-Mills theory, or Yang-Mills theory with adjoint-representation fields, in
any dimension. The Polyakov model falls in the latter class, as do most
theories we discuss in these notes. The modern-day way to phrase this
distinction is to note the presence of a “1-form”353535This symmetry shall be
more carefully defined in Section 3.1, in a way sufficient for our discussion.
Here we note that the so-called “generalized global symmetries,” of which
1-form symmetries are an example, act not on local fields (operators) but on
extended objects, such as Wilson line operators. The most familiar symmetries
usually discussed in QFT courses—those acting on local fields and
operators—are called “0-form symmetries” in this generalized framework. We
refer the reader to [15] for a detailed definition of higher-form symmetries
in the continuum and to [2] for a lattice description of center symmetry. We
note that both the lattice and continuum ways of thinking about the 1-form
center symmetry are quite useful. $Z_{2}$ (for $SU(2)$ gauge group) center
symmetry in theories without dynamical fundamental fields. Center symmetry
will play an important role in our $R^{3}\times S^{1}$ discussion and we shall
introduce the relevant aspects in what follows. For now, we proceed with a
somewhat hand-waving discussion of confinement.

Confinement is probed for by studying the potential energy between probe
static “quarks.” We shall take these to be nondynamical colour sources in the
fundamental representation of the gauge group (one can think of them as the
limit of heavy $m\rightarrow\infty$ dynamical fields). As already mentioned,
recall Exercise 3 and footnote 26, to study the way the dynamics reacts to
nondynamical fundamental probes, one inserts a fundamental Wilson loop in the
path integral of the theory and studies its expectation value.

Figure 5:  The rectangular Wilson loop used to study the potential between
probe fundamental sources.

The Wilson loop along a closed spacetime contour $C$ is defined as

$W[C]=\text{tr}\;{\cal{P}}e^{\;i\oint\limits_{C}A_{\mu}^{a}T^{a}dx^{\mu}}\leavevmode\nobreak\
,$ (43)

where the trace is in the fundamental representation (and so are the
generators $T^{a}$) and ${\cal{P}}$ denotes path ordering along the contour
$C$. That $W[C]$ is gauge invariant follows from the following

> Exercise 7:
>
> Consider the open Wilson line defined as
> ${\cal{P}}e^{\;i\int\limits_{x_{1}}^{x^{2}}dx^{\mu}A_{\mu}^{a}(x)T^{a}}$,
> where the integral is over any path from $x_{1}$ to $x_{2}$. Notice that
> path ordering is defined similar to the usual time ordering appearing in the
> Dyson formula, only with respect to a parameter $\tau$ parameterizing the
> spacetime contour $x^{\mu}(\tau)$. Show363636There are various ways to
> proceed: it helps to first consider $x_{2}$ and $x_{1}$ infinitesimally
> close and then exponentiate the result, or one might write a differential
> equation w.r.t. one of the end points, or consult a textbook proof, say
> [45].  that under gauge transformations, $A_{\mu}\rightarrow
> g(A_{\mu}-i\partial_{\mu})g^{\dagger}$,
>
>
> ${\cal{P}}e^{\;i\int\limits_{x_{1}}^{x^{2}}dx^{\mu}A_{\mu}^{a}(x)T^{a}}\rightarrow
> g(x_{1}){\cal{P}}e^{\;i\int\limits_{x_{1}}^{x^{2}}dx^{\mu}A_{\mu}^{a}(x)T^{a}}g(x_{2})^{\dagger}\leavevmode\nobreak\
> .$ (44)
>
> Then argue that if the path $C$ is closed, so that $g(x_{1})=g(x_{2})$,
> $W[C]$ is invariant.

Taking the contour $C$ to be the one shown on Figure 5, we interpret the
insertion of $W[C]$ as the creation of a quark-antiquark pair, which we let
separate a distance $R$, then propagate for some time $T$, and then annihilate
it. The expectation value of this Wilson loop operator, taking $R$ and $T$ to
infinity, can then be interpreted as giving the potential energy $V(R)$
between the static sources:373737For brevity, the coefficient multiplying the
exponent here is set to unity. In reality, the Wilson loop operator suffers
renormalization, which affects precisely this prefactor (as divergences are
local, they effect the operator defined on the loop $C$, but not the potential
we are interested in). This is already seen in the calculation of $W[C]$ in
QED.

$\langle W[C]\rangle=e^{-V(R)T}\leavevmode\nobreak\ .$ (45)

We say that the theory is confining if, in the infinite $T$ and infinite $R$
limit, $V(R)=\Sigma R$ with $\Sigma$ a proportionality constant of mass
dimension two known as the string tension.383838A general result following
from reflection positivity is that the static potential can not grow faster
than $R$ [46]. The expectation value of $W[C]$, for large loops $C$, is then
said to obey the “area law” $\langle
W[C]\rangle=e^{-\Sigma\text{Area}\;[C]}$.393939Running somewhat ahead, we
shall see below that for theories with dynamical fundamental matter fields, an
area law does not hold due to string breaking. Thus such theories are not
“confining” in the sense discussed here [47], see also [2] .

For confining theories at strong coupling, the area law has been demonstrated
numerically, using lattice gauge theory, or via the analytic strong-coupling
expansion on the lattice. The presence of a linear confining potential
$V(R)=\Sigma R$ can be intuitively explained by the presence of a string, or a
colour-field flux tube whose tension or energy per unit length is $\Sigma$.
The world sheet of the string spans the area of the Wilson loop on Figure 5,
leading to the area law.

In free Maxwell theories, the dependence of the expectation value of $W[C]$ on
$C$, for large contours, can be easily calculated. One finds that $V(R)$ is
the Coulomb potential in the relevant dimension. This result applies to our
perturbatively-free IR theory (19), where $V(R)$ is logarithmic, as
appropriate in 3d.

Our goal in this Section is to show that in the Polyakov model, after taking
into account the monopole-instanton gas, $W[C]$ obeys the area law. To this
end, imagine that we insert a Wilson loop operator $W[C]$ in the path integral
of the $SU(2)$ theory. As we want to study large loops, with size larger than
$v^{-1}$ (as well as $1/m_{\sigma}$), it makes sense to use the IR effective
field theory (38), but to do this, we have to reduce $W[C]$ from (43) to a
form using the relevant IR variables. We do this in two steps: first we reduce
$W[C]$ to the Cartan-subalgebra variables: $W[C]$ of eqn. (43) becomes
$W_{IR}[C]=e^{\;{i\over 2}\oint\limits_{C}A_{\mu}^{3}dx^{\mu}}+e^{\;-{i\over
2}\oint\limits_{C}A_{\mu}^{3}dx^{\mu}}$, i.e. the sum of two $U(1)$ Wilson
loop operators of charges $\pm 1/2$. This is expected, as an $SU(2)$-doublet
quark has two components that have opposite charges under the unbroken $U(1)$.
As our long-distance theory is abelian, we may as well consider separate
insertions of the charges $\pm 1/2$ Wilson loops in the long-distance theory
partition function. Notice, however, that in the $UV$ theory these
contribution come as a sum, as indicated above.

Thus, let us introduce the $U(1)$ Wilson loop of quantized integer charge $q$,
where we dropped the isospin index,

$W_{q}[C]=e^{\;i{q\over 2}\oint\limits_{C}A_{\mu}dx^{\mu}}\leavevmode\nobreak\
.$ (46)

Here, $q=\pm 1$ corresponds to fundamental representation probes, while, e.g.
$q=2$ would be one of the three component of the adjoint (vector)
representation, which has Cartan-$U(1)$ charges $\pm 2$ and $0$. The
definition of $W_{q}[C]$ given on the r.h.s. of (46) is in terms of the
electric variables appropriate to the electric version of the IR theory (19).
As discussed in Section 2.4 (recall Exercise 3), in the dual-photon theory,
the insertion of the operator $W_{q}[C]$ in the path integral is defined by
the prescription to integrate over $\sigma$-field configurations that have a
$2\pi q$ monodromy around the contour $C$.404040This can also be expressed in
a way that we shall not utilize and only mention for completeness. It is,
however, very useful for numerical calculations of string tensions in more
complicated models, for details see [48, 49]. Integrate by parts in the
exponent of (46) to rewrite $W_{q}[C]=\exp(i{q\over 2}\int_{S,\partial
S=C}d^{2}s^{\mu}{1\over 2}\epsilon_{\mu\nu\lambda}F^{\nu\lambda})$, where
$s^{\mu}$ is normal to the surface $S$ spanning $C$. As this form involves
only the gauge invariant field strength, it can be written via
$\partial_{\mu}\sigma$ using the duality relation (21). Then insert $W_{q}$
into the path integral of the dual photon theory to find its expectation
value. At the semiclassical level, one calculates this integral by solving the
classical equation of motion for $\sigma$. One finds that the contribution of
the $W_{q}$ insertion to the equation of motion requires $2\pi q$ jump of
$\sigma$ upon crossing $S$, ensuring correct monodromy around $C$. We also
note that Polyakov used the instanton gas picture to justify a calculation
essentially identical to the one we describe [24, 25], but using the results
of Exercise 6 along the way. Another approach to calculating string tensions,
allowing to smoothly interpolate between the logarithmic Coulomb behaviour of
the quark-antiquark potential at short distances and the confining behaviour
at distances greater than $\ell_{D}$, was developed in [50].

Figure 6:  The charge-$q$ Wilson loop $W_{q}[C]$ requires a $2\pi q$ monodromy
of $\sigma$ around any loop $L$ linked with $C$. To obtain an analytic
expression, we imagine that $C$ is in the $xy$-plane and take it to be
infinitely large, running around the perimeter of the plane.

We can now proceed and calculate the expectation value of the Wilson loop in
the theory (42). All we need to do is compute

$\langle
W_{q}[C]\rangle=Z^{-1}\int{\cal{D}}\sigma\;\text{``}W_{q}[C]{\text{''}}\;e^{\;-{g_{3}^{2}\over(4\pi)^{2}}\int
d^{3}x\left[{1\over
2}(\partial_{\mu}\sigma)^{2}+m_{\sigma}^{2}(1-\cos\sigma)\right]}\leavevmode\nobreak\
,$ (47)

where the insertion of $\text{``}W_{q}[C]{\text{''}}$ indicates, as shown on
Figure 6, that the path integral has to be taken over $\sigma$-field
configurations with $2\pi q$ monodromy around any loop $L$ linked with $C$.
Calculating this path integral precisely is beyond our current ability.
However, we shall now show that a classical field configuration, i.e. a saddle
point of the path integral (47), with the correct monodromy around $C$ exists.
Furthermore, we shall argue that the action of this classical field
configuration is proportional to the area of $C$, for sufficiently large $C$.
Thus, evaluating the path integral (47) in the saddle point approximation
gives rise to the area law

$\langle W_{q}[C]\rangle\sim e^{-S_{class}[\sigma_{q}[C]]}\leavevmode\nobreak\
=e^{-\Sigma_{q}\text{Area}(C)}\leavevmode\nobreak\ .$ (48)

As per footnote 37, calculating the prefactor is both difficult and not of
particular interest to us. The quantity $\Sigma_{q}$ is the string tension of
the confining string, labelled by the charge $q$. In what follows we shall
consider the minimal charge $q=1$, corresponding to a fundamental $SU(2)$
colour source, but shall make some comments about adjoint charges later.

Figure 7:  The domain wall solution (51) plotted for $m_{\sigma}=1$. It is
clear that the thickness of the wall is a few times the Compton wavelength of
the dual photon, as the naive estimate shows.

It should be clear that even the saddle point evaluation of (47) is not easy,
as one needs to analytically find classical $\sigma$-field configurations that
extremize the action in (47) and have the right monodromy, for general
contours $C$. To simplify matters and get an analytic handle, we imagine that
the Wilson loop $C$ is a planar one, encompassing the entire $xy$-plane (i.e.,
we take the loop from Figure 6 to be like the one on Figure 8). In this limit
having a $2\pi$ monodromy around $C$ implies that as one crosses the $z=0$
plane from $z=-\infty$ to $z=+\infty$, the $\sigma$ field should exhibit a
$2\pi$ jump. Further, in the limit when $C$ is very (infinitely) large, we
expect that the classical solution for $\sigma$ will not depend on $x$ and
$y$. Thus, we have reduced the problem to a rather manageable one, namely to
find a solution of the classical equation of motion
$-\partial_{z}^{2}\sigma+m_{\sigma}^{2}\sin\sigma=0$, such that
$\sigma(-\infty)=2\pi$ and $\sigma(+\infty)=0$. Needless to say, this solution
is easy to find analytically, as the equation is easily integrable. It is
known as the domain wall, or (better) kink, solution of the sin-Gordon model.
The analytic form of the domain wall solution is easy to find, see Figure 7,
but it is also instructive to use simple reasoning to find an estimate for its
action.

Recall that the $\sigma=0$ and $\sigma=2\pi$ ground states are identified
(thus, the $\sigma$-field configuration we are looking for is not a true
domain wall, which requires distinct vacuum states on the two sides, but we
shall continue to use this name). First, the domain wall is $x,y$-independent,
so the domain wall action $S_{class.}={g_{3}^{2}\over(4\pi)^{2}}\int
d^{3}x\left[{1\over
2}(\partial_{\mu}\sigma)^{2}+m_{\sigma}^{2}(1-\cos\sigma)\right]$ will be
proportional to the area of the $xy$-plane, which we denote as
$A_{xy}$.414141One can numerically solve for contours of arbitrary shape and
confirm this for very large $C$, see [48, 49] and Figure 9.  Further, as
$z\rightarrow\pm\infty$, the solution ends up in a state where the potential
vanishes. As the $\sigma$ field is massive, these vacuum values at infinity
will be approached exponentially. Thus, we expect that the change
$\delta\sigma=2\pi$ across the domain wall occurs over some finite distance
$\delta z$. Then, an estimate of the action is $g_{3}^{2}A_{xy}(\delta
z({2\pi\over\delta z})^{2}+m_{\sigma}^{2}\delta z)$, dropping numerical
constants. To find the minimal action, extremize w.r.t. $\delta z$ to find
$\delta z\sim m_{\sigma}^{-1}$ and find the action:

$S_{class.}=A_{xy}\;{1\over 2\pi^{2}}\;g_{3}^{2}m_{\sigma}\leavevmode\nobreak\
.$ (49)

Thus, the Wilson loop path integral, evaluated at the saddle point (48),
yields for the fundamental Wilson loop

$W_{1}[C_{xy}]\sim e^{-A_{xy}\Sigma_{1}},\leavevmode\nobreak\ \text{and string
tension}\leavevmode\nobreak\ \Sigma_{1}={1\over
2\pi^{2}}\;g_{3}^{2}m_{\sigma}\leavevmode\nobreak\ .$ (50)

The precise numerical coefficient in (49) can be found in the following

> Exercise 8:
>
> Show that the $z$-dependent domain wall solution with boundary conditions as
> shown on Fig. 7 is
>
> $\sigma(z)=4\arctan e^{-m_{\sigma}z}.$ (51)
>
> Then, compute its action per unit area and show that it is as given in (49).

Equation (49) is the fruit of our long labour through the many steps of this
rather long Section 2. The upshot is that we have shown that large fundamental
Wilson loops in the Polyakov model obey an area law. We have obtained this
result by calculating the path integral (47) of the dual-photon theory in the
saddle point approximation (whose validity requires a large action, and is, a
posteriori, justified in the limit of large area). We have found that the
fundamental string tension is $\Sigma_{1}\sim g_{3}^{2}m_{\sigma}$ and is
determined by the dual photon mass, due to the proliferation of monopole-
instantons.

In addition to having calculated the string tension (50), we can now also
elaborate on the emergence of the confining string by interpreting the result
of the calculation in Minkowski spacetime. Begin by considering (see Figure 8)
the Wilson loop surrounding the $xy$-plane whose expectation value (50) we
computed. The profile of the $\sigma$ field as a function of $z$ is as shown
on Figure 7. Of course, for a finite-size loop, this $z$-only dependence
should hold far away from the edges. This can be verified numerically, see
Figure 9. A detailed numerical analysis shows that the value of the string
tension (49) for infinite separation of the sources (computed using (51)) is
quickly achieved by the string configuration on the figure, already for quark-
antiquark separations of a few (${\cal{O}}(10)$) dual-photon Compton
wavelengths.

Figure 8:  Continuing the Euclidean picture to Minkowski space. Taking $x$ to
be the Euclidean time, we intersect the Wilson loop with a fixed-time plane
(shown by the dotted line) and plot the resulting field configuration on
Figure 9.

Now we imagine taking the $x$ direction to be the Euclidean time and take a
cross section of the picture across the dotted plane (along $yz$) on Figure 8.
The result is shown on Figure 9. The intersections of the dotted plane with
the Wilson loop are the locations of the static quark sources. The
$\sigma$-field configuration is such that $\sigma$ is constant above and below
the $z=0$ plane (the plane of the Wilson loop), and the gradient is only
nonzero in the region indicated by the dotted lines. The width of this region
is proportional to the dual photon Compton wavelength. Now recall that from
the duality (21), the gradient of $\sigma$ is a $90$-degree rotation of the
electric field $\vec{\cal{E}}$. Thus, we conclude that the electric field of
the quark-antiquark pair is zero everywhere except for a flux tube region of
width $1/m_{\sigma}$ connecting the sources. This flux tube is the confining
string, which, of course, is a line, as our space is $R^{2}$.

We find that confinement of colour caused the electric field of the static
sources, instead of representing the dipole picture found in Maxwell
electrodynamics, to become collimated in the flux tube (line) between the two
quarks. The relevant scales we found are the string tension, $\sim
g_{3}^{2}m_{\sigma}$, and the width of the flux tube, $\sim 1/m_{\sigma}$.

Figure 9:  The spatial configuration of static quark/antiquark sources taken
at a finite separation, obtained as explained in Figure 8, after taking $T$ to
infinity. The potential energy density of the $\sigma$ field is shown. Far
from the sources, one can check that the $\sigma$ field profile approaches
that of the analytic solution from Figure 7. The gradient of the $\sigma$
field is nonzero only in a flux tube region of width $1/m_{\sigma}$ connecting
the static sources. In terms of the electric variables, recall the duality
relation (21), the $\vec{\cal{E}}$-field flux of the sources is collimated in
a flux tube (line in $R^{2}$), the semiclassical confining string of the
Polyakov model. (This Figure as well as Figures 19 and 20 are taken from [48,
49] with the sole purpose to qualitatively illustrate the physics, as the
Figures are not always for $SU(N=2)$. The numerical methods used to obtain
them are explained in these references.)

We are now almost ready to move to $R^{3}\times S^{1}$. But before doing this,
let us go back to the beginning of this Section 2.6 and the 1-form $Z_{2}$
center symmetry mentioned there. It was stated there that it is the theory
without fundamental dynamical quarks that enjoys the $Z_{2}$ center symmetry.
Using the picture of confinement that we developed, we can now physically
explain the distinction between theories with $Z_{2}$ center symmetry (the
ones without dynamical fundamental representation quarks) and those without
$Z_{2}$ center symmetry (the ones with finite-mass fundamental representation
quarks).

We used the IR theory of the Polyakov model to find the string tension
confining $q=1$ quark sources, i.e. fundamental representation quarks. But
what about higher charges, for example $q=2$? As explained above, this
includes adjoint representation colour sources. In the IR theory, one can
find, similar to what we did here, configurations composed of two domain walls
that create a $2\pi\times 2$ monodromy of the $\sigma$ field and thus carrying
the necessary flux to confine $q=2$ sources (see Figure 19 in Section 4.2.2).
Thus, one would conclude that $q=2$ colour sources are also confined. However,
$q=2$ is the charge of adjoint fields (the $W$-bosons) under the unbroken
$U(1)$, in our normalization where fundamentals have $q=1$. In the IR theory,
the $W$ bosons were integrated out, as they had mass $m_{W}\sim v$. Now
imagine having a $q=2$ confining flux tube, whose energy grows linearly as
$\Sigma_{2}R$, where $R$ is the separation between the $q=2$ sources. Clearly,
there exists an $R$ sufficiently large so that $\Sigma_{2}R>2m_{W}$. But then,
energetics suggests that it would be advantageous for a $W^{\pm}$ pair to be
created out of the vacuum, as this would lower the energy: the $W^{+}$ would
combine with the $q=-2$ source into an uncharged object, as would the $W^{-}$
with the $q=2$ source. In other words, the pair creation of $W^{\pm}$ would
break the $q=2$ flux tube. The linear potential turns into a constant at large
$R$, as the screened sources do not interact. Thus, confinement, the area law,
and the linear potential have disappeared.

We

learn two lessons from this discussion:

  1. 1.

First, we learn that in the Polyakov model, there is a breakdown of the IR
effective field theory in a $q=2$ flux tube background for asymptotically
large $R$, due to the breaking of the confining flux tubes. This is a
qualitative argument, but there is no reason to doubt its correctness, despite
the fact that calculating the $W^{\pm}$ pair-creation probability in the flux
tube background is a difficult problem.

  2. 2.

Another lesson one can draw from the above is the following. We could repeat
the argument in the theory with dynamical fundamental quarks, no matter how
heavy: a $q=1$ flux tube with tension $\Sigma_{1}$ will, upon increasing $R$,
find it energetically advantageous to break by the pair creation of a heavy
quark-antiquark pair, which will then screen the fundamental sources. Thus, in
the theory with dynamical fundamental charges, the Wilson loop can not have an
area law, as the string connecting fundamental sources is unstable to pair
creation.424242It is clear, then, that there is no clear-cut notion of
confinement in the standard model, which has dynamical fundamental quarks. In
fact, there is no symmetry distinction [47] between the confined and Higgs
phases, see also [2] and, for recent developments, [51].

It would seem, then, that there is some property that distinguishes the theory
with or without dynamical fundamental fields, and that this property is
related to the stability of confining strings. In fact, there is a global
symmetry that acts on flux tubes (which are lines in space, hence the name
“1-form” symmetry, meaning that the symmetry acts on line operators). This
1-form center symmetry ensures the stability of flux tubes. In our $SU(2)$
theory, it is a $Z_{2}$ symmetry. This means that a single flux tube ($q=1$)
is stable, but two flux tubes (combined, two $q=1$ tubes correspond to $q=2$)
are unstable, as explained above. For more formal definitions, see [2] for a
lattice perspective and [15] for a modern “generalized symmetry” perspective.
We shall use and explain in more detail a restricted notion of center symmetry
in Section 3.1. As explained in Section 3.4.3, it will play an important role
in our construction.

Summary of 2.6

: We showed that the fundamental representation Wilson loop in the Polyakov
model obeys an area law. We found that there is a linear confining potential
between fundamental representation static colour sources. We determined the
string tension (50) in terms of the fundamental parameters of the UV $SU(2)$
theory. We also explained the semiclassical nature of the confining flux tube,
see Figure 9, and estimated its parameters. Finally, we qualitatively
discussed the role of the “1-form” $Z_{2}$ center symmetry. This symmetry
distinguishes $SU(2)$ theories with or without dynamical fundamental
representation fields and its presence ensures the stability of confining flux
tubes.

##  3 From $\mathbf{R^{3}}$ to $\mathbf{R^{3}\times S^{1}}$: generalities.

We are now ready to move from $R^{3}$ towards $R^{3}\times S^{1}$. However,
before studying confinement, we have to come to grips with some developments
dating back to the 1980’s and 1990’s. The first is the so-called
“GPY”-potential, found in studies of high-temperature gauge theories [52]. The
second concerns the properties of monopole-instanton solutions in circle
compactified theories discovered simultaneously (and independently) in studies
of quantum field theory [10] and string-theory D-branes [9, 53].

###  3.1 Holonomy, Polyakov loop, center symmetry, and the Weyl chamber.

The first thing we shall do is to “dispose” of the Higgs field $A_{4}^{a}$ of
the Polyakov model (1) as an entity distinct from the gauge field. More
precisely, we shall incorporate it into the theory as an intrinsic part of the
4d $SU(2)$ theory: the component of the gauge connection in the $S^{1}$
direction.

In other words, we shall now reverse the logic around (5), where we used the
4th dimension as a formal tool to study the BPS limit of monopole-instanton
solutions. Explicitly, we now consider a 4d gauge theory with one coordinate
compactified on a circle, i.e. $x^{4}\equiv x^{4}+L$, where $L$ is the $S^{1}$
circumference. We use $M,N$ to denote $R^{4}$ indices, while $\mu,\nu$ are
$R^{3}$ indices; $M=4$ denotes the $S^{1}$ component. The 4d Euclidean
Lagrangian is

$L_{4d}={1\over 4g_{4}^{2}}F_{MN}^{a}F^{MN\;a}\leavevmode\nobreak\ -i\theta
q,$ (52)

and $g_{4}$ denotes the dimensionless 4d gauge coupling taken at some UV scale
to be specified later, $\theta$ is the theta-angle and $q$ is the topological
charge density.434343See eqn (71) and Exercise 11. The topological term will
not be important until the next Section and so we omit it in what follows. We
assume familiarity with the standard construction of $\theta$ vacua in Yang-
Mills theory, see [27].

Without loss of generality,444444This may require a (necessarily short)
explanation. On a general compact manifold, one needs to introduce coordinate
patches and relate the gauge fields on different patches via transition
functions (gauge group elements). $S^{1}$ can be considered as an interval
$[0,L]$ with $A(0)$ and $A(L)$ related by a transition function. One can show
that there exists a gauge transformation making the transition function unity,
implying $A(0)=A(L)$.  the gauge fields can be taken periodic on the $S^{1}$:
$A_{M}^{a}(x_{4}+L)=A_{M}^{a}(x_{4})$. Expanding $A_{M}$ in Fourier (also
called Kaluza-Klein, KK, or Matsubara) modes on the $S^{1}$, it is easy to see
that modes with $x^{4}$-dependence carry momentum quantized in units of $1/L$.

Our studies of the nonperturbative properties of the $R^{3}\times S^{1}$ gauge
theory will, as in the Polyakov model, use a tower of effective theories. As
alluded to in the Introduction, and will be discussed in more detail below, to
ensure calculability, we shall take the highest energy scale in the problem to
be the compactification scale $1/L$. To reduce the 4d theory to one where
excitations of mass $\geq 1/L$ are integrated out, we proceed, at tree-level,
to neglect all $x^{4}$ dependence, taking $\partial_{4}=0$ in (52). Then, we
obtain an effective IR theory valid on $R^{3}$, at energy scales
$\mu\ll{1\over L}$ by integrating the 4d lagrangian $L_{4d}$ over $x^{4}$. The
resulting 3d Lagrangian is:

$L_{3d}={L\over 4g_{4}^{2}}F_{\mu\nu}^{a}F^{\mu\nu\;a}+{L\over
2g_{4}^{2}}(D_{\mu}A_{4})^{a}(D^{\mu}A_{4})^{a}+\ldots.$ (53)

This lagrangian is quite similar to (5). One difference is the replacement
$1/g_{3}^{2}\rightarrow L/g_{4}^{2}$. This is simply the way the 3d
(dimensionful) and 4d (dimensionless) couplings are related upon
compactification, not taking into account quantum corrections. The factor of
$L$ in the numerator occurs because we integrated (52) over the $S^{1}$
coordinate to obtain (53). This is a correct procedure at tree level, but the
question of how the quantum loop corrections affect $L_{3d}$ needs to be
addressed. In particular, in a 4d theory, the coupling $g_{4}^{2}$ runs
logarithmically and one needs to indicate what scale this coupling is taken at
in (53). We shall make this more precise in what follows (although we shall
not delve into all detail, as some of it is inessential for our purposes). At
this point, we appeal to common sense: the natural expectation is that the
relevant scale should be of order $1/L$, since neglecting the Kaluza-Klein
modes means that all excitations heavier than $1/L$ have been integrated out.

The other difference between (5) and $L_{3d}$ of (53) are the “$\ldots$”
terms. These indicate that quantum corrections can generate terms of a form
different from the ones already shown in $L_{3d}$. We can appeal to symmetries
and dimensional analysis to guess what additional terms may occur. The terms
in the $\mu\ll 1/L$ effective action should be gauge invariant, with respect
to 3d gauge transformations, and they should respect the global symmetries of
the 4d theory (as usual, we assume that the theory can be regulated in a way
respecting the symmetries so that the effective action preserves them). Some
of the terms that are allowed to occur in $L_{3d}$ are traces of higher powers
of the 4d gauge invariant field strengths $F_{\mu\nu}$ and $F_{\mu 4}$, with
spacetime indices appropriately contracted. These terms should be suppressed
by inverse powers of $1/L$, the lightest KK-mode mass scale, and are expected
to be irrelevant at $\mu\ll 1/L$ (in our weak-coupling set up).

Other allowed terms may appear less obvious and have the form of a potential
for the $A_{4}$ component. Notice that $A_{4}$ is part of the $SU(2)$
connection and is not invariant under $SU(2)$ gauge transformations, but
transforms as $A_{4}\rightarrow g(A_{4}-i\partial_{4})g^{\dagger}$, where $g$
is an $SU(2)$ group element. A simple, but not quite precise, argument in
favour of the possible appearance of a potential term for $A_{4}$ is that,
from the perspective of the $\mu\ll 1/L$ theory, $A_{4}$ is an adjoint scalar.
This is because the inhomogeneous term in the gauge transformation of $A_{4}$
vanishes if $g$ is taken independent on $x^{4}$.

A better argument, bringing extra insight, is to notice the following. We have
compactified the 4d theory on an $S^{1}$ by integrating out all modes of
energy $\geq 1/L$. Thus, the resulting 3d theory is expected to be local in
3d, but not in the $S^{1}$-coordinate. In particular, there exist an object
that is not local on the $S^{1}$ but is local in $R^{3}$, the Wilson loop
winding around the $S^{1}$. It is also known as the Polyakov loop; we shall
interchangeably call $\Omega$ a Polyakov loop or a winding Wilson loop. The
fundamental representation Polyakov loop is defined as

$\Omega(\vec{x})={\cal{P}}e^{\;i\int\limits_{0}^{L}dx^{4}A_{4}^{a}(\vec{x},x_{4})T^{a}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
\text{det}\Omega=1,\leavevmode\nobreak\ \leavevmode\nobreak\
\Omega^{\dagger}\Omega=1\leavevmode\nobreak\ ,$ (54)

where we stress that $\Omega$ is an $SU(2)$ group element. Sometimes, $\Omega$
is also called the “gauge holonomy” around $S^{1}$. Here ${\cal{P}}$ denotes
path ordering, already discussed near (43) and $T^{a}$ are the fundamental-
representation generators of $SU(2)$. Recalling Exercise 7, from eqn. (44)
there we know that under gauge transformations on $R^{3}\times S^{1}$,
$g(\vec{x},x_{4})\in SU(2)$,

$\Omega(\vec{x})\rightarrow
g(\vec{x},0)\Omega(\vec{x})g(\vec{x},L)^{\dagger}\leavevmode\nobreak\ .$ (55)

This makes it clear that $\Omega(\vec{x})$ transforms as a unitary adjoint
field under $x_{4}$-independent gauge transformations. Further, it is also
clear that the trace of the Polyakov loop $\text{tr}\Omega(\vec{x})$ is
invariant under periodic $SU(2)$ gauge transformations obeying
$g(\vec{x},L)=g(\vec{x},0)$.

There are many properties concerning the Polyakov loop (54) that are relevant
to our future discussion and that we shall unravel as we go on. We first go
back to enumerating the possible gauge invariant terms that can be added to
our $L_{3d}$ of (53). The above discussion implies that we can add the
following, local in $R^{3}$ but not local on $S^{1}$, gauge invariant terms to
(53):454545Note that the gauge invariants $\text{tr}(\Omega^{k})$ are not
independent and can be expressed through $(\text{tr}\Omega)^{k}$ via the
characteristic equation of the matrix $\Omega$:
$\Omega^{2}-\Omega\;\text{tr}\Omega+\text{det}\Omega=0,$ (56) which is easily
verified in our $2\times 2$ case.

$\ldots\supset\sum\limits_{k,p\in Z}{c_{kp}\over
L^{3}}\;(\text{tr}\;\Omega)^{k}\;(\text{tr}\;\Omega^{\dagger})^{p}\leavevmode\nobreak\
,$ (57)

where we wrote the $L^{-3}$ factor based on dimensional analysis and $c_{kp}$
are unknown dimensionless coefficients (clearly, they should obey relations
ensuring the reality of the effective action terms (57)). The origin and
physical consequences of adding terms like (57) to (53) will be discussed
later (the sum over integer powers $k$ and $p$ will be seen to arise due to
integration out of KK modes).

Now we reveal one property that distinguishes the terms with $k+p$-even in
(57). The $k+p$-even terms are invariant under the following $Z_{2}$
transformation

$Z_{2}^{(1)}:\;\text{tr}\;\Omega\rightarrow-\text{tr}\;\Omega\leavevmode\nobreak\
$ (58)

while terms with odd $k+p$ violate (58). We shall use the notation
$Z_{2}^{(1)}$ for (58), in some departure from the literature where it is
sometimes called “0-form center symmetry.” This is because from the point of