# Jena lectures on generalized global symmetries: principles and applications

Nabil Iqbal<EMAIL_ADDRESS>Centre for Particle Theory, Department
of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK

###### Abstract

This is an elementary set of lectures on generalized global symmetries
originally given at the Jena TPI School on QFT and Holography, designed to be
accessible to the reader with a basic knowledge of quantum field theory.
Topics covered include an introduction to higher-form symmetries with selected
applications: Abelian and non-Abelian gauge theories in the continuum and the
lattice, statistical mechanical systems, the Adler-Bell-Jackiw anomaly from
the point of view of non-invertible symmetry, and magnetohydrodynamics.
Mathematical formalism is kept to a minimum and an emphasis is placed on
understanding the global symmetry structure present in well-known models.

###### Contents

  1. I Introduction
    1. I.1 Motivation: what is gauge theory?
  2. II Ordinary global symmetries, phases of matter, and the Landau paradigm
    1. II.1 Ordinary global symmetries
    2. II.2 The old Landau paradigm
    3. II.3 Physics beyond the Landau paradigm
    4. II.4 Symmetries as topological operators
  3. III Higher form symmetries
    1. III.1 Idea
    2. III.2 Phases of 1-form symmetry
    3. III.3 Genuine and non-genuine operators
      1. III.3.1 Non-genuine local operators: 2d compact boson
  4. IV Applications
    1. IV.1 Free electromagnetism
    2. IV.2 Interlude: Goldstone’s theorem
      1. IV.2.1 Goldstone’s theorem for 0-form symmetries
      2. IV.2.2 Goldstone’s theorem for higher form symmetries
    3. IV.3 Electrodynamics coupled to matter
    4. IV.4 Gauge theory: $SU(N)$
      1. IV.4.1 SU(N) gauge theory in the continuum
      2. IV.4.2 SU(N) gauge theory on the lattice
      3. IV.4.3 Confinement and the 1-form symmetry
    5. IV.5 Gauge theory: $\mathbb{Z}_{2}$
      1. IV.5.1 Kramers-Wannier duality and the Ising spin model
      2. IV.5.2 Higgs/confinement continuity
    6. IV.6 Effective description of spontaneously broken $\mathbb{Z}_{N}$ 1-form symmetry: topological order
      1. IV.6.1 BF theory
      2. IV.6.2 Ground states in a topologically ordered phase
  5. V Non-invertible symmetries: the axial anomaly
    1. V.1 Different types of anomalies
    2. V.2 Charge defect operators for $U(1)_{A}$
  6. VI More applied applications
    1. VI.1 Finite temperature: static phenomena
    2. VI.2 Finite temperature: magnetohydrodynamics
      1. VI.2.1 Lightning review of ordinary hydrodynamics
      2. VI.2.2 Magnetohydrodynamics

##  I Introduction

One of our most basic tools in the quantitative understanding of nature is the
idea of symmetry. Any reader who has opened these lecture notes probably does
not need to be convinced that symmetries are important. This hypothetical
reader might, however, be surprised to know that in recent years the very idea
of symmetry in quantum field theory has undergone something of a renaissance.
As we will discuss in detail below, it turns out that the symmetries that one
learns about in an introductory physics course are actually only a very
special case of a much richer structure. This richer structure is sometimes
called that of generalized global symmetries Gaiotto:2014kfa .

This new set of ideas captures very basic physical principles. It is not
complicated or overly formal (at least, not more so than most quantum field
theory already is), and it seems to the author that it could very easily be
taught in elementary quantum field theory courses. It is the goal of these
lecture notes – based on a set of lectures first given at the Jena TPI School
on QFT and Holography – to present the material at such an introductory level,
and thus to equip a reader who already has a working knowledge of quantum
field theory with a similar working toolkit of these new symmetries.

The field is vast and growing rapidly, and these notes do not attempt to
provide an authoritative overview. Instead they necessarily discuss a rather
narrow and somewhat biased sampling of the ideas, centered around explaining
how these generalized global symmetries exist already in many of the simplest
and most familiar systems – e.g. ordinary electrodynamics, or pure Yang-Mills
theory, or even the 2d Ising model – that theoretical physicists study.
Several other recent reviews discussing different aspects of generalized
symmetries include Cordova:2022ruw ; Brennan:2023mmt ; Gomes:2023ahz ;
Shao:2023gho ; Schafer-Nameki:2023jdn ; Bhardwaj:2023kri . Our discussion is
philosophically aligned with the point of view of McGreevy:2022oyu .

###  I.1 Motivation: what is gauge theory?

Before beginning, let us start with a question. In most elementary treatments
of field theory, the concept of gauge theory plays a central role.

What, then, is a gauge theory?

At first, this seems to be a silly question: clearly, a gauge theory is a
theory with a gauge field. For example, in three spacetime dimensions, we
could study free electrodynamics, with action

$S_{\mathrm{gauge\;theory}}[A]=\frac{1}{4g^{2}}\int d^{3}xF_{ij}F^{ij}\qquad
F_{ij}=\partial_{i}A_{j}-\partial_{j}A_{i}$ (I.1)

This is obviously a gauge theory.

However, in three dimensions, it is a well-known fact that one can dualize it,
i.e. rewrite this gauge theory in terms of a massless scalar field ${\theta}$
using the following change of variables:

$\frac{1}{g^{2}}F_{ij}={\epsilon}_{ijk}\partial^{k}\theta$ (I.2)

and then write down an action describing the system in terms of the new scalar
field as

$S_{\mathrm{gauge\;theory?}}[{\theta}]=\frac{g^{2}}{2}\int
d^{3}x\;(\partial_{i}\theta)(\partial^{i}\theta)\ .$ (I.3)

The theory described by (I.3) is precisely equivalent to (I.1), but simply
expressed in different variables111This is the mapping between symmetry-broken
phases of the two quantum field theories – a charged scalar field and the
Abelian-Higgs model – which are related by particle-vortex duality
Peskin:1977kp ; dasgupta1981phase ; see e.g. Chapter 8 of tong2018gauge  for a
review..

Now it seems that it would be ridiculous to claim that a free massless scalar
in three dimensions is a gauge theory; yet if (I.3) is truly the same as
(I.1), then apparently it is.

This problem is not restricted to this example, and actually sits at the heart
of many of the most fascinating phenomena in theoretical physics. Some other
systems which are not obviously gauge theories – but which nevertheless can be
usefully described by Lagrangians involving gauge fields – include systems as
disparate as (the right density of) electrons in a magnetic field (i.e. the
fractional quantum hall effect), or quantum gravity on AdS${}_{5}\times S^{5}$
(i.e. the AdS/CFT correspondence Maldacena:1997re ). The issue here is that a
gauge symmetry is not really a symmetry at all – it is more precisely stated
as a redundancy, a feature of the variables that we choose to describe the
system, rather than an intrinsic property of the system itself.

But if gauge fields can come and go in this way when we change the variables
that we use, then what is the point of gauge theory? A potential answer to
this question is that gauge theory is intimately associated with describing
extended structures – e.g. in the most familiar case of electrodynamics, the
extended objects in question are electric and magnetic field lines. We should
then seek an organizing framework built around the existence of such extended
objects. Generalized global symmetries – as we will see in detail below –
provide precisely such a framework.

Exercise Prove that the quantum theories defined by (I.1) and (I.3) are
equivalent, by placing the action (I.1) in a path integral and performing an
appropriate change of variables.

##  II Ordinary global symmetries, phases of matter, and the Landau paradigm

These lectures will be about new sorts of global symmetries. Let’s start by
carefully understanding ordinary global symmetries.

###  II.1 Ordinary global symmetries

To orient ourselves, let’s consider a field theory with a $U(1)$ global
symmetry. For much of what we will discuss in this section, you can consider
any field theory you like, but for concreteness let us consider a charged
scalar field:

$S[\phi]=\int
d^{d}x\left(\partial\phi^{\dagger}\partial\phi+V(\phi^{\dagger}\phi)\right)$
(II.1)

We will assume that the potential has at least a mass term, and possibly some
self interactions:

$V(\phi^{\dagger}\phi)=m^{2}\phi^{\dagger}\phi+\frac{\lambda}{4}(\phi^{\dagger}\phi)^{2}+\cdots$
(II.2)

In almost all of these lectures we will work in Euclidean spacetime with $d$
dimensions, where $d$ will take various interesting values as we proceed.

This is a classical action, but we can make it into a quantum theory by
inserting its exponential into a path integral:

$Z=\int[d\phi]\exp(-S[\phi,\phi^{\dagger}])$ (II.3)

Now we know very well that this theory has a $U(1)$ global symmetry. What does
it act on? This is a question with an obvious answer: it acts on the
fundamental degree of freedom, the $\phi$ field, as

$\phi(x)\to\phi^{\prime}(x)=e^{i\alpha}\phi(x)$ (II.4)

where $e^{i\alpha}\in U(1)$ is a space-time independent phase, and clearly the
action is invariant. $\phi(x)$ is a local operator. Now through the usual
Noether procedure we can construct a current which has one index for this
symmetry:

$j^{\mu}(x)=i(\phi^{\dagger}\partial^{\mu}\phi-\phi\partial^{\mu}\phi^{\dagger})$
(II.5)

The current is conserved on-shell. Classically, this just means that if $\phi$
satisfies the equations of motion we have

$\partial_{\mu}j^{\mu}=0\ .$ (II.6)

This classical conservation equation is somewhat modified in the quantum
theory. If we apply Noether’s theorem inside the path integral, then (as
explained in any introductory quantum field theory textbook), we find an
expression of the form:

$\langle\partial_{\mu}j^{\mu}(x)\phi(y)\cdots\rangle=i\delta^{(d)}(x-y)\langle\phi(y)\cdots\rangle$
(II.7)

In other words: the current is conserved up to delta function terms at the
location of the insertion of charged operators. This is the Ward identity, and
one should view this expression as telling us what it means to be charged
operator.

Finally, global symmetries allow us to define a conserved charge. Normally we
do this in Lorentzian signature in the following way:

$Q(t)=\int_{\mathbb{R}^{d-1}}d^{3}xj^{0}(t,\vec{x})$ (II.8)

where we do the integral over a time-slice $\mathbb{R}^{d-1}$. Let’s just
verify why this is conserved:

$\frac{d}{dt}Q(t)=\int_{\mathbb{R}^{d-1}}d^{3}x\partial_{t}j^{0}(t,\vec{x})=-\int_{\mathbb{R}^{d-1}}d^{3}x\partial_{i}j^{i}(t,\vec{x})=\mbox{boundary
term}=0$ (II.9)

Ler us draw a picture of this equation in Figure 1: imagine that there are
some particles sitting around. They trace out worldlines in space-time. The
world-lines can’t end, because the particle number is conserved. This integral
over the time-slice intersects all of the world lines – you can move it up and
down in time and get the same answer, which is just the count of the particle
number. A key point here is that ordinary symmetries are associated with
counting particles.

Figure 1: Particle world-lines can’t end: no matter at what time-slice we
evaluate $Q(t)$ on we get the same number.

Note also that the local field $\phi(x)$ is the thing that creates a particle
or anti-particle. Another point is the charged object creates the thing that
the charge counts.

One more piece of formalism is that it is often useful to couple in an
external gauge field $a_{\mu}$ to the current $j_{\mu}$. Here this coupling is
implemented by demanding that the system is invariant under the combined
transformation:

$a\to a+d\Lambda\qquad\phi\to\phi e^{i\Lambda}$ (II.10)

where now $\Lambda(x)$ is a function of spacetime. To do this, we have to
replace the normal derivative $\partial_{\mu}\phi$ with the gauge-covariant
one $D_{\mu}\phi=\partial_{\mu}\phi-ia_{\mu}\phi$, and we can consider the
action:

$S[\phi;a]=\int
d^{d}x\left((D\phi)^{\dagger}D\phi+V(\phi^{\dagger}\phi)\right)\qquad
Z[a]=\int[d\phi]\exp(-S[\phi;a])$ (II.11)

Note that $a$ is a fixed external field, and the invariances above mean that
the path integral is invariant under external gauge transformations of $a$:

$Z[a+d\Lambda]=Z[a]$ (II.12)

Furthermore, we can compute the current by constructing
$j^{\mu}(x)=\frac{\delta Z[a]}{\delta a_{\mu}(x)}$.

Now, what are global symmetries good for?

###  II.2 The old Landau paradigm

One very important use of global symmetries is to classify the phases of
matter. Note that the theory described above has two different phases, which
we now study in turn:

  1. 1.

The Unbroken Phase. Here we have $m^{2}>0$, and we have a unique vacuum in
which the field $\phi$ is basically zero, i.e. $\langle\phi(x)\rangle=0$.
Because this vacuum is invariant under the $U(1)$ symmetry, we say that the
$U(1)$ symmetry is unbroken.

In this phase we can imagine doing a calculation of the following two-point
function:

$\langle\phi^{\dagger}(x)\phi(0)\rangle\sim\exp(-m|x|)\ .$ (II.13)

If we turn off the interactions, this is again a standard computation in
elementary field theory. Physically, the reason that this decays exponentially
in space is that $\phi(0)$ creates a particle at the origin which must then
survive until it is annihilated at $\phi^{\dagger}(x)$. This costs action
because the particle is heavy, and thus the exponential decay.

  2. 2.

The Spontaneously Broken Phase. Here we have $m^{2}<0$, and the potential
(II.2) has a family of minima at
$\phi^{\dagger}\phi=\frac{{\lambda}}{2m^{2}}\equiv v^{2}$. Thus the vacuum
will have a nonzero value of $\langle\phi\rangle=v$. In this phase we say that
the particles have condensed, and the $U(1)$ symmetry is spontaneously broken.

We can parametrize the interesting degree of freedom here by writing

$\phi(x)=v(x)e^{i\theta(x)}$ (II.14)

describing the modulus and phase of the field respectively. What can we say
about the dynamics in the broken phase? Now plugging this into (II.1) we see
that the field $v(x)$ acquires a mass which pins it at the minimum of the
potential. Thus at low energies we may ignore it. However the phase field
$\theta(x)$ is massless, so the leading low-energy action is

$S=\int d^{d}x\;v^{2}(\partial\theta)^{2}+\cdots$ (II.15)

$\theta$ is the Goldstone mode of the symmetry. Its masslessness is much
stronger than the classical argument we just gave: Goldstone’s theorem states
non-perturbatively that if there is a spontaneously broken symmetry (as
defined by $\langle\phi\rangle\neq 0$) then there is necessarily a gapless
mode in the spectrum. We will prove this again shortly as a warm-up for
fancier things.

Let us consider another way to describe the broken symmetry. Consider
computing

$\lim_{|x|\to\infty}\langle\phi^{\dagger}(x)\phi(0)\rangle\sim\langle\phi^{\dagger}(x)\rangle\langle\phi(0)\rangle=v^{2}$
(II.16)

Here the first equality follows from cluster decomposition, i.e. the fact that
things far away should be independent. Thus an equivalent way to state that
the symmetry is spontaneously broken is to note that the two-point function of
the charged field saturates and factorizes, becoming independent of the
separation, as contrasted with the exponential decay shown in (II.13). In this
form this is sometimes called off-diagonal long-range order.

We say that $\phi$ is the order parameter for the transition, because it
behaves so differently in the two phases.

Now, basically when you have a symmetry, these are the two phases that you can
have: unbroken and spontaneously broken. You can also ask what happens right
between the two: there you have a phase transition:

  1. 3.

Phase Transition: in the model above, the boundary between the two phases is
at $m^{2}=0$. It turns out that at such a continuous transition we then have a
conformal field theory which is characterized by the symmetry group in
question (in this case $U(1)$) and the number of dimensions. It seems
reasonable to describe the transition by the action we wrote down earlier, but
setting $m^{2}=0$:

$S=\int
d^{d}x\left(\partial\phi^{\dagger}\partial\phi+\frac{\lambda}{4}(\phi^{\dagger}\phi)^{2}+\cdots\right)$
(II.17)

If we now compute the correlation function of the charged operator we will
find something which is in between the two behaviors (i.e. constant and
exponential) shown above: it is a power law.

$\langle\phi^{\dagger}(x)\phi(0)\rangle\sim\frac{1}{|x|^{2\Delta}}\,,$ (II.18)

where $\Delta$ is a number that is interesting to calculate – it is called an
operator dimension in CFT or a critical exponent in statistical physics. As it
turns out, for $d<4$ this number depends on the interactions in an important
way.

Note that from this point of view, the reason why a CFT exists is to describe
the universal physics present at the phase transition between two distinct
phases.

Now the point of all of this is that it gives us a way to describe phases of
matter. A phase of matter is a classification of the long-distance physics (as
captured e.g. by the correlation functions (II.13) and (II.16)), and in many
cases it is determined completely by how the symmetries are realized.

Let us study some more examples:

  1. 1.

$U(1)$ superfluid: if we imagine putting many atoms in a box, then they have a
conserved particle number enforced by a $U(1)$ symmetry much like the one we
just described222The effective theory is somewhat different because the system
is non-relativistic, and the background value of the charge density $\langle
j^{0}\rangle$ is nonzero.. Here if the symmetry is unbroken, then we call it
the normal phase. On the other hand, if you cool the system down enough (see
e.g. ketterle1999experimental  for a discussion of the experimental
situation), then the effective potential for the “field that creates an atom”
develops a minimum away from the origin, and this field condenses, breaking
the $U(1)$ number symmetry. This is now called a superfluid.

  2. 2.

Magnetization: the simplest way to think about this is to imagine a bunch of
spins $\vec{s}$ that are fixed on the sites of a lattice. There can be a non-
Abelian rotational333You may be worried that in real life this should be a
spacetime symmetry and not an internal symmetry. This is correct and manifests
in spin-orbit couplings which can couple the spin to the orbital angular
momentum of atoms. Often this can be neglected, or we work on explicit lattice
systems where it is absent from the construction. symmetry that acts on the
spins as $\vec{s}\to R\vec{s}$. We often consider very simplified settings –
e.g. you could imagine a simpler symmetry group $\mathbb{Z}_{2}$ that simply
acts on a scalar spin $s$ as $s\to\pm s$, which defines what is called the
Ising model (we will review this model in Section IV.5.1). If this rotational
symmetry is unbroken, then the system is called a paramagnet, and if it
spontaneously broken – i.e. if all the spins are lined up – we call it a
ferromagnet.

  3. 3.

Liquids and solids: imagine have a collection of atoms moving freely;
sometimes they can stop moving and form a crystal, in which case translational
symmetry is spontaneously broken down to some discrete subgroup (e.g. for a
square lattice it would be $\mathbb{Z}^{d}$). A solid is thus a phase where
translational symmetry is spontaneously broken, and a liquid is a phase where
translational symmetry is unbroken.

What about the difference between liquid and gas? These have the same
symmetries, and thus do not actually define different phases – this means that
we can connect them continuously in a phase diagram, as is the case e.g. for
water.

A further point here is that the transitions between phases can be continuous
or discontinuous (i.e. first order). If they are first order there is not much
more we can say; but if they are continuous then we can write down an
effective theory involving a coarse-grained order parameter to describe the
transition basically as we did in (II.17).

These ideas – that the phases of matter are classified by patterns of
spontaneously broken and unbroken symmetry, and that the transitions between
them can be described by the symmetries alone – are called the Landau paradigm
of the phases of matter. It is extremely successful and is one of the
cornerstones of our understanding of nature.

It also seems that it doesn’t always work.

###  II.3 Physics beyond the Landau paradigm

Here are some phases which are outside the simplest Landau paradigm:

  1. 1.

Gauge theory: it seems that gauge theories can come in different phases. For
$U(1)$ gauge theory in the continuum we have the Coulomb and Higgs phases,
which seem different. What is the order parameter? There is no local order
parameter for this, and similarly for non-Abelian gauge theory and
confinement.

  2. 2.

Topological order: sometimes there are phases of matter which interact in an
interesting manner with the topology of the space where they are defined –
e.g. if you place a density of electrons in a strong magnetic field, you get
the fractional quantum Hall effect, which basically says that the electron
somehow splits into three smaller objects with fractional charge. These three
pieces of the electron do interesting things if you move them around each
other. Fascinatingly, the ground state degeneracy of this phase is different
if you put it on a torus or a sphere.

  3. 3.

Gravity: gravity can be thought of as a phase of matter with a gaplessly
dispersing mode. It does not appear to fit into the conventional Landau
paradigm; we will not discuss this in these lectures, but some recent work in
this direction appears in Hinterbichler:2022agn ; Benedetti:2021lxj ;
Cheung:2024ypq .

The goal of these lectures is to generalize our understanding of global
symmetries in two directions: higher-form global symmetries, and non-
invertible global symmetries. We will see that once we do this, we will
actually be able to significantly extend the Landau paradigm in a way that
encompasses some of the phases above. We stress that there are many other ways
to go beyond the Landau paradigm: for an overview on how generalized
symmetries interact with them, see McGreevy:2022oyu .

###  II.4 Symmetries as topological operators

To do this, we first need to obtain a refined understanding of ordinary
symmetries. Note that the description above made heavy use of the fundamental
field $\phi(x)$: but from a modern point of view, in an age of duality where
the notion of “fundamental” has little meaning, this seems quaint. We seek a
notion of global symmetry that does not require any particular choice of
fundamental fields, and works in a duality-invariant way.

So, let us take a step back: if we have a symmetry, we have a conserved
current:

$\partial_{\mu}j^{\mu}=0\qquad d\star j=0$ (II.19)

Here we will start using a differential forms notation. Now let’s consider the
following operation: consider a codimension-$1$ manifold
${{\mathcal{M}}}_{d-1}$ and form the following charge integral:

$Q({{\mathcal{M}}}_{d-1})=\int_{{{\mathcal{M}}}_{d-1}}d^{d-1}x\sqrt{h}\;n_{\mu}j^{\mu}=\int_{{{\mathcal{M}}}_{d-1}}\star
j$ (II.20)

This appears to be a function of the manifold ${{\mathcal{M}}}_{d-1}$ –
however because the current $j^{\mu}$ is conserved, it is actually independent
under small wiggles of ${{\mathcal{M}}}_{d-1}$: it is a topological defect in
the theory. The notion of topological defect will appear multiple times, so
here it is good to make an informal definition:

A topological defect on some submanifold ${{\mathcal{M}}}$ is something that
does not change under small deformations of ${{\mathcal{M}}}$.

Note that it is exactly the same sort of integral that we do when constructing
a usual conserved charge, except that we now take ${{\mathcal{M}}}_{d-1}$ to
be a possibly wiggly manifold rather than a flat time slice
$\mathbb{R}^{d-1}$.

Figure 2: Integrating divergenceless current $\star j$ on a codimension-1
manifold ${{\mathcal{M}}}_{d-1}$.

Thus whenever we have a normal symmetry, we obtain a codimension-$1$
topological defect. This simple idea now supports many generalizations, which
we will now explore.

What happens when you consider the interaction of this charged defect with a
charged operator ${{\mathcal{O}}}(x)$? In the presence of a charged operator,
we know from the Ward identity that the current is not conserved, i.e. we have

$\partial_{\mu}j^{\mu}(x){{\mathcal{O}}}(y)=i\delta^{(d)}(x-y){{\mathcal{O}}}(y)$
(II.21)

(where it is understood that this expression holds in the path integral). Thus
if we try to drag the topological defect ${{\mathcal{M}}}_{d-1}$ through the
insertion of a charged local operator, it not topological anymore; instead the
operator picks up a phase, i.e. as in Figure 3

$\displaystyle\left(Q({{\mathcal{M}}}_{d-1}^{1})-Q({{\mathcal{M}}}_{d-1}^{2})\right){{\mathcal{O}}}(x)$
$\displaystyle=Q({{\mathcal{M}}}^{\prime}_{d-1}){{\mathcal{O}}}(x)=\oint_{{{\mathcal{M}}}^{\prime}_{d-1}}\star
j{{\mathcal{O}}}(x)$ $\displaystyle=\int_{B_{d}}(d\star
j){{\mathcal{O}}}(x)=i\int_{B_{d}}\delta(x-y){{\mathcal{O}}}(x)=i{{\mathcal{O}}}(x)$
(II.22)

where above $B_{d}$ “fills in” ${{\mathcal{M}}}^{\prime}_{d-1}$, i.e.
$\partial B_{d}={{\mathcal{M}}}^{\prime}_{d-1}$.

Figure 3: Deforming ${{\mathcal{M}}}^{1}$ to ${{\mathcal{M}}}^{2}$ receives a
contribution only from the insertion of the local operator
${{\mathcal{O}}}(x)$.

There’s another way to say this. If we formulate the charge operator on an
$S^{d-1}$ that wraps the operator and collapse it onto the operator as a
point, we find:

$Q(S^{d-1}){{\mathcal{O}}}(x)=i{{\mathcal{O}}}(x)$ (II.23)

It is also convenient to consider the topological defect that constructs a
finite charge rotation by an angle $\alpha$. As expected this is the
exponential of the infinitesimal charge rotation, so we have:

$U_{{\alpha}}({{\mathcal{M}}}_{d-1})\equiv\exp(i{\alpha}Q({{\mathcal{M}}}_{d-1}))\qquad
U_{{\alpha}}(S^{d-1}){{\mathcal{O}}}(x)=e^{i\alpha}{{\mathcal{O}}}(x)$ (II.24)

i.e. when we collapse the charge generator onto the local operator it acts as
a linear phase rotation on the field.

Figure 4: Collapsing the charge generator $U_{{\alpha}}(S^{d-1})$ onto
${{\mathcal{O}}}(x)$ acts as a phase rotation.

Note also that if you imagine composing two of these charge operators by
forming them on parallel manifolds and fusing them together we find:

$U_{{\alpha}}({{\mathcal{M}}}_{d-1})U_{\beta}({{\mathcal{M}}}_{d-1})=U_{\alpha+\beta}({{\mathcal{M}}}_{d-1})$
(II.25)

Here the last expression ${\alpha}+\beta$ should be understood in the spirit
of group composition of the two $U(1)$ group elements $e^{i{\alpha}}$ and
$e^{i\beta}$.

Figure 5: Fusing together two charge defects results $U_{{\alpha}}$ and
$U_{\beta}$ results in a combined charge defect $U_{{\alpha}+\beta}$.

We may now generalize this somewhat to an arbitrary non-Abelian global
symmetry. Consider a symmetry group $G$, and imagine that we have fields
${{\mathcal{O}}}_{a}$ that transform in some representation $r$ under $G$, so
that for every $g\in G$ there are matrices $V^{a}_{b}(g)$ that provide a
linear action of the group, so

$V^{a}_{b}(g)V^{b}_{c}(g^{\prime})=V^{a}_{c}(gg^{\prime})$ (II.26)

The analogue of the above is now that for every element $g$ of the symmetry
group $G$ we have topological defect operators $U_{g}({{\mathcal{M}}}_{d-1})$.
They can be composed in the obvious way as follows:

$U_{g}({{\mathcal{M}}}_{d-1})U_{g^{\prime}}({{\mathcal{M}}}_{d-1})=U_{gg^{\prime}}({{\mathcal{M}}}_{d-1})$
(II.27)

To understand the action on local operators ${{\mathcal{O}}}_{a}(x)$, we
imagine forming a defect on an $S^{d-1}$ and collapsing it around the operator
as in (II.24):

$U_{g}(S^{d-1}){{\mathcal{O}}}_{b}(x)=V^{a}_{b}(g){{\mathcal{O}}}_{a}(x)$
(II.28)

We stress that here the $V$’s are just linear finite-dimensional matrices, and
this relation shows how the quantum operator $U$ acts as a finite dimensional
rotation on the charged degrees of freedom. So in the $U(1)$ case e.g. we have
that $r$ is a one-dimensional representation and the $V$ is just a phase
$e^{i\alpha}$. For $SU(2)$ we might take $r$ to be in the fundamental and then
the $V$’s would be exponentials of the Pauli matrices, and so on.

This may seem unfamiliar, but it is simply a restatement of the regular setup
of global symmetries.

Exercise 1. Prove, by cutting open the path integral on a time-slice, that
(II.23) is equivalent to the familiar operator equation
$[Q,{{\mathcal{O}}}(\vec{x})]=i{{\mathcal{O}}}(\vec{x})$ (II.29) where here
$Q$ is defined in the conventional way as in (II.8). 2. Derive the
corresponding operator relation for the finite non-Abelian transformation from
(II.28).

Finally, we should note that essentially the same formalism works for discrete
symmetries. If we have a theory with a discrete symmetry group $G$, then we do
not have a conserved current for it, but we do have a set of topological
operators that enact the action of the discrete symmetry as in (II.28). As an
example, imagine a real scalar field theory invariant under the
$\mathbb{Z}_{2}$ symmetry $\phi\to-\phi$; then there would exist a topological
defect $U$ which has the following action when collapsed on $\phi$:

$U_{\mathbb{Z}_{2}}(S^{d-1})\phi(x)=-\phi(x)$ (II.30)

As there is only a single element in $\mathbb{Z}_{2}$ and it is its own
inverse, we have the simple fusion rule:

$U_{\mathbb{Z}_{2}}({{\mathcal{M}}}_{d-1})U_{\mathbb{Z}_{2}}({{\mathcal{M}}}_{d-1})=1$
(II.31)

The main point of all of this is just to note that symmetries give us
topological operators, often called charge defects.

Because the charged objects here are $0$-dimensional local operators, ordinary
global symmetries are called $0$-form symmetries.

##  III Higher form symmetries

Once we realize that global symmetries essentially define a topological sector
of a theory, we can generalize these in many productive ways. We will first
consider higher-form symmetriesGaiotto:2014kfa , which basically ask the
question: if normal symmetries are associated with the conservation of
particles, what symmetries are associated with the conservation of extended
objects, such as strings or branes? This is not a question only for string
theorists: such objects exist in real life.

###  III.1 Idea

We will start with the continuous case. Consider an antisymmetric two-index
current $J^{\mu\nu}=J^{[\mu\nu]}$ which is conserved:

$\partial_{\mu}J^{\mu\nu}=0\qquad d\star J=0$ (III.1)

Just as ordinary 0-form global symmetries count particles, this object is
associated with a $U(1)$ symmetry which counts strings! This is called a
$1$-form global symmetry.

At some rudimentary level the extra index tells us which way the string
points. Let’s dig into this a little bit; first, following (II.20) we can form
the following object

$Q({{\mathcal{M}}}_{d-2})=\int_{{{\mathcal{M}}}_{d-2}}\sqrt{h}n^{\mu\nu}J_{\mu\nu}=\int_{{{\mathcal{M}}}_{d-2}}\star
J$ (III.2)

which we integrate over a codimension-$2$ manifold. For the same reasons as
above, this is a topological defect, which is now codimension-$2$. This is
2-dimensional in 4d. For intuition, imagine a sea of strings – i.e.
2-dimensional worldsheets without end – poking through the screen on which you
read this, as in Figure 6. To count all of them we don’t need to integrate
over the whole room, we just integrate over the screen – we can then move the
surface ${{\mathcal{M}}}_{d-2}$ upwards in time or forwards in space and get
the same answer, because the strings have no end.

Figure 6: To count 2d string worldsheets, one does an integral over a
codimension-$2$ surface, not a codimension-$1$ surface as for particle
worldlines; compare this to Figure 1.

We have formulated ordinary symmetries in a fancy way above so that we can
exploit the fact that much of the formalism carries over very simply. First,
what is the charged object? Recall from before that a charged object is the
“thing that creates the thing that you count”, in other words it should be the
object who creates a string.

This is a line operator $W(C)$ defined on a 1d curve $C\subset\mathbb{R}^{d}$,
which can be imagined as creating a string living along that curve $C$ at an
instant in time. If it has charge $q$ then the Ward identity in this case now
becomes:

$\partial_{\mu}J^{\mu\nu}(x)W(C)=iq\int_{C}ds\frac{dX_{C}^{\nu}}{ds}\delta^{(d)}(x-X_{C}(s))W(C)$
(III.3)

which is a somewhat clunky way to write that the current is not conserved
whenever we evaluate it on the worldline of the extended charged operator.
Note that the extra index $\nu$ comes from the geometry of the line operator.

We can write this much more elegantly if we define the $(d-p)$ form-valued
delta function $\delta_{C_{p}}(x)$ which depends on a $p$-submanifold $C_{p}$:
it is zero whenever $x\notin C_{p}$ and “integrates to 1”, which in this case
means that

$\int_{\mathbb{R}^{d}}B_{p}\wedge\delta_{C_{p}}(x)=\int_{C_{p}}B_{p}$ (III.4)

for all test forms $B_{p}$. The case above is for $p=1$, and we can write it
as

$d\star J(x)W(C)=iq\delta_{C}(x)W(C)$ (III.5)

This is just notation to replace the integral in (III.5).

Now again we can construct the object who generates the finite rotation by an
angle $\alpha$:

$U_{{\alpha}}({{\mathcal{M}}}_{d-2})=\exp(i{\alpha}Q({{\mathcal{M}}}_{d-2}))$
(III.6)

We should now ask what happens if we drag this object through a line operator
$W(C)$. The idea is the same as in (II.22) – $W(C)$ will pick up a phase. This
is most cleanly represented if we imagine placing the defect on a $S^{d-2}$
that links the $W(C)$ and then collapsing it, as in Figure 7.

$U_{{\alpha}}(S^{d-2})W(C)=e^{iq\alpha}W(C)$ (III.7)

Figure 7: Collapsing the 1-form charge defect $U_{{\alpha}}(S^{d-2})$ onto a
line operator acts as a phase rotation on the line.

Exercise: Prove (III.7) from (III.5).

This is the idea of a 1-form symmetry.

It can clearly be generalized to any $p$-form symmetry, which in the
continuous case means that there exists a $p+1$-form symmetry current
$J^{\mu_{1}\cdots\mu_{p+1}}$, charge defects $Q({{\mathcal{M}}}_{d-p-1})$, and
charged surface operators that exist on submanifolds $C_{p}$ of dimensionality
$p$. A 0-form symmetry counts particles, a 1-form symmetry strings, a 2-form
symmetry bedsheet-type objects, and so on.444In fact the idea can even
usefully be generalized in the other direction to a (-1)-form symmetry
Aloni:2024jpb , at least in the $U(1)$ case, though the notion of “counting”
things becomes a bit abstract.

It cannot however be generalized to the non-Abelian case: to understand this
let’s consider doing the group composition (II.27) for a $p$-form symmetry.

$U_{g}({{\mathcal{M}}}_{d-p-1})U_{g^{\prime}}({{\mathcal{M}}}_{d-p-1})=U_{gg^{\prime}}({{\mathcal{M}}}_{d-p-1})$
(III.8)

The ordering on the right hand side of $gg^{\prime}$ was inherited from the
ordering on the left hand side. For codimension $1$ objects (i.e. for $p=0$)
there is a well-defined notion of ordering – this is the time-ordering that is
inherited from the path integral. However for higher codimension objects there
isn’t – you can always move one around the other. Thus the ordering is
ambiguous and the right hand side of the fusion cannot depend on it, and thus
$g$ and $g^{\prime}$ necessarily commute.

We can however easily have discrete higher-form symmetries – e.g.
$\mathbb{Z}_{N}$ – which will indeed play an important role below. Just as for
0-form symmetries, we then have a charge operator
$U_{g}({{\mathcal{M}}}_{d-p-1})$ where $g\in\mathbb{Z}_{N}$. See the recent
review Brennan:2023mmt  for an extensive discussion of discrete higher-form
symmetries.

###  III.2 Phases of 1-form symmetry

Let us now understand the phases of 1-form symmetries. We will do this as we
did around (II.13) and (II.16), i.e. by understanding the behavior of the
expectation value of the charged object $\langle W(C)\rangle$:

  1. 1.

Unbroken phase: in the unbroken phase, we have that for large closed curves
$C$ there is the area law:

$\langle W(C)\rangle\sim\exp\left(-T\mbox{Area}[C]\right)$ (III.9)

where $\mathrm{Area}[C]$ is the area of the minimal surface that fills in the
curve $C$, i.e. the minimal surface $S$ such that

$\partial S=C\ .$ (III.10)

This is the higher-form analogue of the exponential decay shown in (II.13).
The way to understand this is that the objects that we count – i.e. the string
worldsheets – are massive with tension $T$; thus the line operator creates a
2d worldsheet which must fill in the curve.

You will be familiar with the area law as a diagnostic for confinement in non-
Abelian gauge theory; we will return to this below.

  2. 2.

Spontaneously broken phase: here you should imagine that the strings have
“condensed”, i.e. that the tension has dropped to zero. What does this mean?
This may seem alarming; however in this case it means that the behavior of the
line operator has changed to a perimeter law.

$\langle W(C)\rangle\sim\exp\left(-mL[C]\right)$ (III.11)

where $L[C]$ is just the length of the curve (or more generally any local
functional of the curve data). The idea here is that just as in (II.16) the
correlation function has factorized as much as possible; however the line
operator can always be redefined by a c-number which is a local functional of
$C$, and thus the behavior above is essentially the same as saying that the
line operator is a constant which is independent of $C$. The parameter $m$ is
thus non-universal.

Finally, we could imagine the transition between these phases, where we would
have

  1. 3.

Phase transition: if the transition is continuous, we would expect to see a
conformal field theory describing it, and here we would have

$\langle W(C)\rangle=\exp\left(-m^{\prime}L[C]\right)f[C]$ (III.12)

where $f[C]$ is expected to depend on $C$ in a nontrivial manner, and might
capture some interesting universal data determined by the transition.
Unfortunately there are not too many continuous transitions involving the
spontaneous breaking of a 1-form symmetry; they turn out to usually be first-
order555For a possible explanation see Iqbal:2021rkn .. An important exception
is $\mathbb{Z}_{2}$ gauge theory in 3d, which we will discuss.

The analogy with 0-form symmetries should be clear. As we will see, this
classification can also be useful – this will provide a useful classification
of phases, and we can do all of the same sorts of things that we did for
0-form symmetries for their higher-form cousins.

Figure 8: Area vs. perimeter laws in the spontaneous breaking of a 1-form
symmetry.

It should be clear how the ideas above generalize to an arbitrary $p>1$.

###  III.3 Genuine and non-genuine operators

At this point we have introduced already a few types of extended objects. This
is a good time to define some more terminology.

A topological operator (which can be either local or an extended defect) is an
object which is invariant under small wiggles of its support. An example is
the identity operator 1 and any of the extended symmetry defects defined
above. (Indeed, by Schur’s lemma, any local topological operator is
necessarily proportional to the identity).

A genuine operator (which can be either local or extended) is an operator
which does not have anything tied to it. Examples are most of the operators
you know and love.

A non-genuine operator (which can be either local or extended) is an operator
who does have something tied to it. Examples include:

  1. 1.

Fields that are charged under gauge symmetries – e.g. in QED with a charged
scalar $\Phi$ necessarily has a Wilson line attached to it, so it is not a
genuine local operator: we have to write it as $\Phi(x)e^{i\int_{x}A}$ where
the Wilson line defines a curve $C$ that ends on $x$. Note that in this
example the Wilson line is usually not topological, which means that you are
generally aware that it is there.

  2. 2.

A more insidious example of this phenomenon is when the “something tied” to
the operator is topological. In that case one might miss it and then be
tempted to conflate a genuine operator with a non-genuine one, leading to
confusion.

  3. 3.

Both cases can happen for extended objects: e.g. you can have a 1d line
operator with a 2d surface tied to it, where the surface might or might not be
topological.

####  III.3.1 Non-genuine local operators: 2d compact boson

Let us study an example of a non-genuine local operator. This section requires
some familiarity with 2d CFT. Consider the compact boson CFT with action

$\frac{1}{2g^{2}}\int d^{2}x\;(\partial\theta)^{2}$ (III.13)

where ${\theta}\sim{\theta}+2\pi$. Let us formulate it on
$S^{1}\times\mathbb{R}$. In that case there is the well-known state operator
correspondence that maps states on the cylinder to local operators on
$\mathbb{R}^{2}$. In particular, the vacuum on the cylinder maps onto the
identity on $\mathbb{R}^{2}$.

Now we are free to formulate boundary conditions as we like on the $S^{1}$. It
is a reasonable thing to demand that ${\theta}$ wind an integer number of
times around the $S^{1}$, i.e. that

${\theta}(\phi)={\theta}(\phi+2\pi)+2\pi n\qquad n\in\mathbb{Z}$ (III.14)

The vacuum under this boundary condition defines a new state $|n\rangle$. We
can then map the vacuum with this boundary condition back to the plane by the
state-operator correspondence. This defines a family of local operators
$V_{n}(x)$ labeled by an integer $n$. These are genuine operators.

One could have formulated it directly on the plane by demanding that in a path
integral formulation of the theory, the integral of the 1-form $d\theta$ on a
circle wrapping the point $x$ be quantized as $2\pi n$, i.e.

$\oint d\theta=2\pi n$ (III.15)

Note that in this formulation we have an operator $V_{n}(x)$ who is defined
not as a polynomial of fundamental fields in the Lagrangian (which is how we
normally define fields) but rather as a boundary condition on fields in the
path integral. This is an example of a disorder operator, which is still
genuine.

Now consider the following wrong-feeling operation: let’s consider a boundary
condition on the $S^{1}$ so that $\theta$ has a non-integer winding:

${\theta}(\phi)={\theta}(\phi+2\pi)+2\pi\alpha\qquad\alpha\notin\mathbb{Z}$
(III.16)

We are free to do this; but what does it mean? Local physics on the cylinder
proceeds mostly as normal; but if we map back to the plane the putative local
operator that we try to define $\tilde{V}_{\alpha}(x)$ will have a branch cut
emanating out of it. This branch cut is the location of the seam where the
field ${\theta}$ jumps by $2\pi{\alpha}$. It is a topological line, as clearly
we can move it around mostly as we like. Thus the operator dual to the non-
integer winding vacuum has a tail – i.e. a topological line operator ending on
it – and is a non-genuine operator.

Exercise: Show that the “tail” above is actually the 1d charge defect for the
0-form $U(1)$ global symmetry associated with ${\theta}$ translation. Hint:
what is the action of the tail on the operator $e^{im{\theta}}$?

##  IV Applications

This has all been rather formal. We will now study a few examples and see what
this formalism can do for us.

###  IV.1 Free electromagnetism

Consider familiar free $U(1)$ electromagnetism in 4d with no matter.

$S[A]=\int d^{4}x\left(\frac{1}{4e^{2}}F^{2}\right)\qquad F=dA$ (IV.1)

The equation of motion is $\partial_{\mu}F^{\mu\nu}=0$. The action above does
not fully specify the theory, as it describes only the Lie algebra of the
gauge group and not the gauge group itself. It will soon be important that we
are talking about compact electromagnetism with gauge group $U(1)$ and not
$\mathbb{R}$. This means that gauge transformations are labeled by elements
$g=e^{i\Lambda(x)}\in U(1)$, with $A\to g^{-1}(A+d)g$.

What are the symmetries? Introductory textbooks would now talk about gauge
symmetry.

However, a sophisticated reader might make the argument that gauge “symmetry”
is a misleading name; gauge symmetry is not really a symmetry at all, but
rather a redundant way of describing the dynamics. As we have discussed above,
it is entirely possible to reformulate a theory described with a Lagrangian
involving gauge fields in terms of a different Lagrangian with no gauge fields
at all. So we can (a tad pretentiously) ask: what is the invariant physical
information present in this theory? In other words, what are the global
symmetries?

This theory has two 1-form global symmetries: an electric one $U(1)^{(1)}_{e}$
and a magnetic one $U(1)^{(1)}_{m}$ with the following currents:

$J^{\mu\nu}_{e}=\frac{1}{e^{2}}F^{\mu\nu}\qquad
J^{\mu\nu}_{m}=\frac{1}{4\pi}{\epsilon}^{\mu\nu\rho\sigma}F_{\rho\sigma}$
(IV.2)

or in forms

$J_{e}=\frac{1}{e^{2}}F\qquad J_{m}=\frac{1}{2\pi}\star F$ (IV.3)

The conservation of $J^{\mu\nu}_{e}$ follows from the equations of motion and
is true only on-shell. This captures the idea that electric field lines are
strings that don’t end (there is no electrically charged matter). The
conservation of $J^{\mu\nu}_{m}$ follows from the Bianchi identity
${\epsilon}^{\mu\nu\rho\sigma}\partial_{\nu}F_{\rho\sigma}=0$, and captures
the idea that magnetic field lines are strings that don’t end (there are no
magnetic monopoles).

What are the charged line operators?

The line operator charged under $U(1)^{(1)}_{e}$ is the Wilson line, i.e. the
worldline of a test electric charge:

$W_{q}(C)=\exp\left(iq\int_{C}A\right)$ (IV.4)

In the presence of the Wilson line we obtain the expression:

$(d\star J_{e}(x))W_{q}(C)=iq\delta_{C}(x)W_{q}(C)$ (IV.5)

These charges are integers $q\in\mathbb{Z}$.

The action of $U(1)^{(1)}_{e}$ on the dynamical degrees of freedom can be
easily understood in this case – they correspond to the shift of $A$ by a
closed 1-form:

$A\to A+\Lambda\qquad d\Lambda=0$ (IV.6)

which leaves the Lagrangian invariant. Under this the Wilson line operator
transforms as a phase

$W_{q}(C)\to e^{iq\int_{C}\Lambda}W_{q}(C)\ .$ (IV.7)

How do we couple an external source $b_{\mu\nu}$ to this 1-form symmetry? In
the electric case this takes the following form:

$S[A;b_{e}]=\int d^{4}x\left(-\frac{1}{4e^{2}}(dA-b_{e})\right)$ (IV.8)

and the 1-form invariance is now enhanced to:

$A\to A+\Lambda\qquad b_{e}\to
b_{e}+d\Lambda\qquad\Lambda\;\;\mbox{arbitrary}$ (IV.9)

This is the 1-form analogue of the external coupling to the source that we did
back in (II.10).

We should note that it is often the case that we cannot identify such a simple
action on the degrees of freedom. For example, the line operator charged under
$U(1)^{(1)}_{m}$ is called an ’t Hooft line, which we will denote by
$T_{p}(C)$– it is the worldline of a test magnetic monopole. It is an example
of a so-called disorder operator Kapustin:2005py , and is defined by demanding
that in the path integral over fields, every $S^{2}$ wrapping the curve $C$
has:

$\oint_{S^{2}(C)}F=2\pi p$ (IV.10)

i.e. we have a magnetic monopole. Note that naively we have $F=dA$, which
clearly means that the integral on the right-hand side of $F$ must vanish; of
course this means that $A$ is not well-defined in the presence of a monopole,
and must be defined patch-wise and sewn together on overlapping patches by
gauge transformations. The consistency of this sewing together with the
integer quantized electric charges in (IV.4) enforces the Dirac quantization
condition,

$qp=\mathbb{Z}$ (IV.11)

which you will remind yourself of in the exercise below. (IV.10) is equivalent
to the following Ward identity for the magnetic current:

$(d\star J_{m}(x))T_{q}(C)=iq\delta_{C}(x)T_{q}(C)$ (IV.12)

For the magnetic symmetry there is no analogue of (IV.6).

Exercise: 1. Prove the Dirac quantization condition in compact $U(1)$ gauge
theory (IV.11). 2. Prove the Ward identities (IV.5) and (IV.12).

Exercise: 1. Couple an external two-form source $b_{m}$ to Maxwell theory for
the magnetic 1-form symmetry $U(1)^{(1)}_{m}$, just as we did for the electric
1-form symmetry in (IV.8). 2. Use this to construct a partition function for
Maxwell theory $Z[b_{e},b_{m}]$ that is invariant under both 1-form shifts
$b_{e}\to b_{e}+d\Lambda_{e},b_{m}\to b_{m}+d\Lambda_{m}$. 3. Actually, this
is impossible. This is a manifestation of a mixed ’t Hooft anomaly between the
two 1-form symmetries. Develop the theory of this anomaly and explain to what
extent each current is conserved.

Now that we have understood that we have two 1-form symmmetries
$U(1)_{e}^{(1)}$ and $U(1)_{m}^{(1)}$, we should understand how they are
realized: i.e. are they spontaneously broken, or unbroken, etc? To do that we
compute the expectation values $\langle W_{q}(C)\rangle$ and $\langle
T_{p}(C)\rangle$.

The calculation of $\langle W_{q}(C)\rangle$ for a circular loop is a textbook
exercise. Heuristically speaking the Wilson line emits photons but they don’t
do too much, and the leading contribution comes from a short-distance
singularity along the perimeter of the loop, so we find:

$\langle W_{q}(C)\rangle\sim\exp(-\Lambda L[C])$ (IV.13)

with $\Lambda$ a non-universal cutoff. This is a perimeter law. $\langle
T_{q}(C)\rangle$ behaves the same way – one way to understand this is that the
action of free electrodynamics is invariant under electric-magnetic duality,
which interchanges the Wilson and ’t Hooft lines. Thus in the normal phase of
“free EM” we see that both higher form symmetries are spontaneously broken.

This is an important fact – the way to think about it that the vacuum is a sea
of “condensed” strings, where the condensed objects are magnetic and electric
field lines. Long-wavelength wiggles in this sea of condensed strings are what
we call photons. Indeed this allows us to ask the question: why is the 4d
photon massless?

When first learning QFT we often hear some words about why the photon is
massless due to gauge invariance, which sounds pretty good until the next
chapter where we learn about the Higgs mechanism, which writes down a
perfectly gauge-invariant action that gives the photon a mass. Thus the
conventional way of thinking about this is somewhat unfulfilling.

The modern way of thinking about this is that the photon is the Goldstone mode
of a spontaneously broken higher-form global symmetry666See Kovner:1992pu  for
a very early discussion of this idea.. We should thus prove Goldstone’s
theorem for higher-form symmetries.

###  IV.2 Interlude: Goldstone’s theorem

Before proving Goldstone’s theorem for higher-form symmetries, we will first
revisit the proof for ordinary 0-form symmetries. The canonical non-
perturbative proof of Goldstone’s theorem involves Hamiltonian methods
Goldstone:1962es ; here we will discuss a Euclidean path-integral proof,
following Hofman:2018lfz .

####  IV.2.1 Goldstone’s theorem for 0-form symmetries

Let us begin by considering the Ward identity for a charged operator
${{\mathcal{O}}}(x)$ at the origin.

$\partial_{\mu}j^{\mu}(x){{\mathcal{O}}}(0)=iq\delta^{(d)}(x){{\mathcal{O}}}(0)$
(IV.14)

Let us now integrate $x$ on both sides over a ball $B^{d}$ which is the
interior of a $S^{d-1}$ of radius $R$ centered at the origin, as in Figure 9.

Figure 9: Geometry used for proof of 0-form Goldstone theorem.

We find using the divergence theorem that

$\oint_{S^{d-1}(R)}d^{d-1}{\theta}\;n_{i}j^{i}(\theta){{\mathcal{O}}}(0)=iq\int_{B^{d}(R)}d^{d}x\delta^{(d)}(x){{\mathcal{O}}}(0)$
(IV.15)

where ${\theta}$ describes the angles parametrizing the $S^{d-1}$. Now the
right hand side integral is trivial and picks up the delta function at the
origin, giving us

$\oint_{S^{d-1}(R)}d^{d-1}{\theta}\;n_{i}j^{i}(\theta){{\mathcal{O}}}(0)=iq{{\mathcal{O}}}(0)$
(IV.16)

So far this has been true in any theory, independent of the phase. Now take
the expectation value of both sides. We find

$\bigg{\langle}\oint_{S^{d-1}(R)}d^{d-1}{\theta}\;n_{i}j^{i}(\theta){{\mathcal{O}}}(0)\bigg{\rangle}=iq\langle{{\mathcal{O}}}\rangle\
.$ (IV.17)

There are now two possibilities:

  1. 1.

The symmetry is unbroken, so $\langle{{\mathcal{O}}}\rangle=0$. The left-hand
side also vanishes. In this case we can say nothing else.

  2. 2.

The symmetry is spontaneously broken, so $\langle{{\mathcal{O}}}\rangle\neq
0$. In this case the left-hand side is both nonzero and independent of the
radius $R$. Thus by spherical symmetry we have that the correlation function
between the the operator at the origin and the current at a distance $R$ from
it satisfies:

$\langle
j^{i}(\theta){{\mathcal{O}}}(0)\rangle\sim\frac{iq\langle{{\mathcal{O}}}\rangle
n^{i}}{R^{d-1}}$ (IV.18)

(Amusingly, this is exactly how we derive the classic $r^{-2}$ potential using
Gauss’s law in elementary EM, which is not a coincidence). Thus there is a
power-law in distance correlation in the theory, and there must be at least
one gapless mode to generate this power-law correlation. This gapless mode is
the Goldstone mode.

This is the fastest proof of the ordinary Goldstone theorem, though to our
knowledge it was first written down in Hofman:2018lfz . Though it is
formulated in Euclidean language, it is conceptually equivalent to the
ordinary Hamiltonian proof; in particular if you cut open the path integral
then you find the commutators that one normally uses.

####  IV.2.2 Goldstone’s theorem for higher form symmetries

We now present the proof for higher form symmetries. It is really exactly the
same as above.777For a proof using Hamiltonian methods, see Lake:2018dqm . We
begin by considering the Ward identity for a $p$-form symmetry:

$d\star J(x)W(C)=iq\delta_{C}(x)W(C)$ (IV.19)

Let us now assume that we are in a spontaneously broken phase, as that is the
case of interest. In that case $W(C)$ obeys a perimeter law as in (III.11)

$\langle W(C)\rangle\sim\exp(-m\mbox{Perimeter}[C])$ (IV.20)

where here “perimeter” denotes the $p$-volume of the submanifold $C$. Note
that as this is a local functional of the geometric data characterizing $C$,
we can define a new charged operator $\overline{W}(C)$ that strips this off,

$\overline{W}(C)\equiv\exp(+m\mbox{Perimeter}[C])W(C)$ (IV.21)

$\overline{W}(C)$ satisfies the same Ward identity as $W(C)$, i.e.

$d\star J(x)\overline{W}(C)=iq\delta_{C}(x)\overline{W}(C)$ (IV.22)

Now just as above, we take $C$ to be an infinite flat $p$ plane and consider a
$(d-p)$ dimensional ball $B^{d-p}$ that intersects $C$ at a single point. This
ball has a boundary $S^{d-p-1}$ and the perpendicular distance from the
intersection point to the $S^{d-p-1}$ is $R$, as in Figure 10.

Figure 10: Geometry used for proof of p-form Goldstone theorem.

Now we integrate $x$ on both sides of (IV.22) over the $B^{d-p}$. On the left
hand side we have

$\int_{B^{d-p}}d\star J(x)\overline{W}(C)=\oint_{S^{d-p}(R)}\star
J(\theta)\overline{W}(C)\ .$ (IV.23)

where $\theta$ is a point on the $S^{d-p}$. On the right hand side the delta
function contributes from a single point, so we have

$\oint_{S^{d-p}(R)}\star J(\theta)\overline{W}(C)=iq\overline{W}(C)$ (IV.24)

Finally we take the expectation value:

$\big{\langle}\oint_{S^{d-p}(R)}\star
J(\theta)\overline{W}(C)\big{\rangle}=iq\langle\overline{W}(C)\rangle$ (IV.25)

By construction $\langle\overline{W}(C)\rangle$ is nonzero, and thus the left
hand side is both nonzero and independent of the radius $R$, which means that
the correlation function must behave as

$\langle\star J(\theta)\overline{W}(C)\rangle\sim\frac{iq}{R^{d-p-1}}$ (IV.26)

i.e. there is a power-law correlation between the current at a distance $R$
from the surface operator. This power law correlation means that there is a
single gapless mode in the spectrum, which is the Goldstone mode.

One can now ask what the low-energy theory is that describes this Goldstone
mode, i.e. what is the higher-form analogue of (II.15)? It will not surprise
you that this is given by a $p$-form gauge theory, i.e.

$S=\int d^{d}x\left(v^{2}(dB_{p})^{2}\right)$ (IV.27)

where the extended charge operator is realized in terms of this degree of
freedom by

$W(C)=\exp\left(iq\int_{C}B_{p}\right)$ (IV.28)

In the case $p=1$ this is exactly the structure of ordinary electromagnetism!
If you run the argument with $J_{e}$ then the low-lying Goldstone mode that
you get is the ordinary photon $B_{p=1}=A$ and (IV.28) is exactly the ordinary
Wilson line. If you run the argument with $J_{m}$ then the Goldstone mode
turns out to be the magnetic photon $B_{p=1}=\tilde{A}$, which is related to
the ordinary one by electric-magnetic duality; then (IV.28) turns out to
represent the ’t Hooft line instead.

Exercise: Prove in the first non-trivial case $p=1$ that the low-energy
effective action (IV.27) saturates the spontaneously broken Ward identity
(IV.26).

This is maybe the first case where we see that the structures from ordinary
symmetries really do carry over to the higher-form case; we really obtain
nontrivial dynamical information, i.e. from the behavior of a certain line
operator we can now state that there exists a gapless mode in the
spectrum.888See Hidaka:2020ucc  for a discussion on how the counting of
higher-form Goldstone modes can be modified when they mix with other fields
that break spacetime symmetries.

###  IV.3 Electrodynamics coupled to matter

Now that we understand that the free photon is the Goldstone mode of a
spontaneously broken 1-form symmetry, we should think about what happens if we
couple in electric matter. Let’s start with a scalar of integer charge $q$,
i.e. the action is now:

$S[\phi,A]=\int
d^{4}x\left(\frac{1}{4e^{2}}F^{2}+(D\phi)^{\dagger}D\phi+V(\phi^{\dagger}\phi)\right)$
(IV.29)

with the usual gauge-covariant derivative $D_{\mu}\phi=\partial_{\mu}\phi-
iqA_{\mu}\phi$ and where we take

$V(\phi^{\dagger}\phi)=m^{2}\phi^{\dagger}\phi+\frac{\lambda}{4}(\phi^{\dagger}\phi)^{2}+\cdots$
(IV.30)

Let us now ask what has happened to the symmetry currents we identified
before. We begin by studying the case $q=1$. The equations of motion now say
that

$\partial_{\mu}J^{\mu\nu}_{e}=iq\left(\phi^{\dagger}D^{\mu}\phi-\phi
D^{\mu}\phi^{\dagger}\right)=j^{\mu}_{\mathrm{el}}$ (IV.31)

i.e. the electric 1-form symmetry $U(1)^{(1)}_{e}$ has been explicitly broken
as the corresponding current is no longer conserved. Physically, the point is
that electric field lines can now end on free electric charges.

There is another way to see that $U(1)^{(1)}_{e}$ is now broken. The charged
Wilson line operators can now end on scalar field insertions as
$\phi(x)\exp\left(i\int_{x}^{\infty}dsA\right)$. The line operator then cannot
have a nontrivial linking with any topological charge operator as we can just
pull the charge operator off the end and take it away.

The magnetic 1-form symmetry $U(1)^{(1)}_{m}$, whose current is:

$J^{\mu\nu}_{m}={1\over 2}{\epsilon}^{\mu\nu\rho\sigma}F_{\rho\sigma}$ (IV.32)

remains, as there are no dynamical magnetic monopoles in this theory; of
course it may still be spontaneously broken. To understand this, we should
separate the two infrared phases of this gauge theory:

  1. 1.

Coulomb phase: this is when the mass in the potential $m^{2}>0$, and so the
scalar $\phi$ is uncondensed. In that case if we go to very long distances we
can essentially ignore the $\phi$ field completely, as the right hand side of
(IV.31) is energetically suppressed. We are then back in the situation of free
EM. Note that the electric 1-form symmetry looks like it has come back – it is
emergent at long distances, and then spontaneously broken. This sort of
situation – i.e. the emergence of a symmetry at long distances in the absence
of fine-tuning – is perhaps somewhat rare for 0-form symmetries but is generic
for 1-form symmetries. The magnetic 1-form symmetry is also spontaneously
broken.

  2. 2.

Higgs phase: here we imagine adjusting the potential by taking $m^{2}<0$ so
that the gauge-charged scalar condenses: this is then the Higgs or
superconducting phase.

Then it’s convenient to write $\phi(x)=v(x)e^{i\theta(x)}$ and the low-energy
physics is described by:

$S=\int
d^{4}x\left(\frac{1}{4e^{2}}F^{2}+v_{0}^{2}(d\theta-A)^{2}+(dv)^{2}+m_{H}^{2}(v-v_{0})^{2}+\cdots\right)$
(IV.33)

It is well-known that in this Higgs phase the spectrum is completely gapped.
What has happened to our 1-form symmetries? The electric one is completely
broken; in this phase you definitely cannot ignore the right hand side of
(IV.31).

To understand how the magnetic symmetry is realized, we should compute
$\langle T(C)\rangle$ in this phase. We are thus asking: what happens if we
place a probe magnetic monopole into a Higgs phase? In the Coulomb phase the
magnetic field lines can spread out. In the Higgs phase, however, they cannot
– the very point of the Higgs phase is that magnetic fields are expelled. If
you force flux in nevertheless, the magnetic field lines are forced into a
tight little bundle called the Abrikosov flux tube.

Let us briefly describe the field configuration of the flux tube: we are
searching for a solution to the equations of motion of (IV.33) that supports
nonzero magnetic flux. It’s helpful to go to cylindrical coordinates

$ds^{2}=d\tau^{2}+dz^{2}+dr^{2}+r^{2}d\phi^{2}$ (IV.34)

and imagine that we are trying to put a nonzero magnetic field in
$F_{r\theta}$. We then find the following configuration. The phase of the
scalar field will wind once around:

$\theta(r\to\infty)=\phi$ (IV.35)

This is terribly singular at the origin, and thus the scalar field will thus
necessarily vanish there and interpolate back to its background value at
infinity.

$v(r\to 0)\approx 0\qquad v(r\to\infty)=v_{0}\qquad\theta(r\to\infty)=\phi$
(IV.36)

This interpolation will take place over a scale set by the Higgs mass
$m_{H}^{-1}$. Finally, because the action contains a kinetic term like
$(\partial_{\phi}\theta-A_{\phi})^{2}$, at infinity $A_{\phi}(r)$ must track
$\theta$:

$\qquad A_{\phi}(r\to\infty)=1$ (IV.37)

Thus the amount of magnetic flux stored in the tube is

$\int_{\mathbb{R}^{2}}F=\oint_{S^{1}(\infty)}d\phi A_{\phi}=2\pi$ (IV.38)

and there is thus a single unit of flux.

Note that this quantization can be traced back to the fact that the field
$\theta$ is winding; by creating an integer winding ${\theta}=\mathbb{Z}\phi$
one can create configurations supporting an integer number of units of flux,
but there is no way within this phase to create (e.g.) half a unit.

This configuration is a 2d worldsheet that fills all of $(\tau,z)$ and is
localized at $r=0$; it has an tension $t_{p}$ per unit worldsheet area which
can be calculated numerically and is nonzero.

Finally, let us return to our basic question: what happens if we insert a ’t
Hooft line into this theory? The ’t Hooft line is a magnetic monopole, which
is the endpoint of an Abrikosov vortex – in other words it will create
magnetic flux which is then forced into the 2d worldsheet described above,
which will result in an area law. We thus have

$\langle T(C)\rangle\sim\exp(-t_{p}\mbox{Area}[C])$ (IV.39)

and the 1-form symmetry in the Higgs phase is unbroken.

This may all seem backwards – isn’t the gauge symmetry spontaneously broken in
the Higgs phase, or something like that? The issue here is really that the
previous statement – though illuminating in the limit of weak gauge coupling
$e\to 0$ – does not really make sense nonperturbatively, because gauge
symmetry is not a symmetry. (For example: what would the order parameter be
for its breaking? There is nothing observable that is charged under the gauge
symmetry, and thus the very idea of an order parameter does not make sense).

We can summarize the situation in the following table:

| $U(1)^{(1)}_{e}$ | $U(1)^{(1)}_{m}$
---|---|---
Free electrodynamics | Spontaneously broken | Spontaneously broken
Massive electrically charged matter | Explicitly broken (emergent in IR) | Spontaneously broken
Higgs phase | Explicitly broken | Unbroken
Massive magnetically charged matter | Spontaneously broken | Explicitly broken (emergent in IR)
Confined phase | Unbroken | Explicitly broken

Table 1: Realization of 1-form symmetries in different phases of 4d $U(1)$
gauge theory

.

To understand the last lines, note that in 4d $U(1)$ electrodynamics we can
imagine also coupling in dynamical magnetic monopoles. If they condense then
we would end up with a confined phase, where it is the Wilson line who has an
area law.

Finally, we can consider a non-trivial minimal charge $q>1\in\mathbb{Z}$. In
this the electric 1-form symmetry is explicitly broken down to
$\mathbb{Z}_{q}^{(1)}$; the idea is that electric flux tubes with flux less
than the minimal charge can’t be screened and so their number is conserved mod
$q$.

###  IV.4 Gauge theory: $SU(N)$

We now turn to non-Abelian gauge theory.

####  IV.4.1 SU(N) gauge theory in the continuum

We will begin by thinking about pure $SU(N)$ gauge theory. The continuum
action depends only on the Lie algebra $\mathfrak{su}(N)$

$S[A]=\frac{1}{2g^{2}}\int d^{4}x\mathop{\rm Tr}(F_{\mu\nu}F^{\mu\nu})$
(IV.40)

where the gauge potentials $A_{\mu}$ and field strength $F_{\mu\nu}$ are both
valued in $\mathfrak{su}(N)$. As it turns out, the structure of allowed line
operators depends intricately not only on the Lie algebra, but also on the
global form of the gauge group – e.g. $SU(N)$ and $SU(N)/\mathbb{Z}_{N}$ are
different groups with the same algebra, and result in corresponding Yang-Mills
theories with the same action that are nevertheless subtly different
Aharony:2013hda 999In fact this sort of ambiguity is present even in the very
familiar Standard Model Tong:2017oea ..

It turns out that any pure Yang-Mills theory with gauge group $G$ has (at
least) a 1-form symmetry given by the center of the gauge group. Recall that
the center $Z(G)$ of a group $G$ is the set of elements that commute with all
elements of $G$, i.e.

$Z(G)=\\{g\in G\mid gg^{\prime}=g^{\prime}g\;\;\mbox{for every }g^{\prime}\in
G\\}$ (IV.41)

For concreteness, in these notes we will only discuss pure $SU(N)$ gauge
theory. For $SU(N)$, we have that $Z(G)=\mathbb{Z}_{N}$, written as

$\begin{pmatrix}e^{\frac{2\pi ik}{N}}&&&\\\ &e^{\frac{2\pi ik}{N}}&&\\\
&&\cdots&\\\ &&&e^{\frac{2\pi ik}{N}}\\\ \end{pmatrix}\qquad
k\in\\{0,\cdots(N-1)\\}$ (IV.42)

which clearly commutes with everything and has unit determinant.

Thus the statement is that pure $SU(N)$ gauge theory has a $\mathbb{Z}_{N}$
1-form symmetry.

To understand this 1-form symmetry, we have to do two things: understand the
charged line operators, and understand the existence of the topological
2-dimensional surface operator.

The line operators are – unsurprisingly – Wilson lines:

$W_{r}(C)=\mathrm{\mathop{\rm
Tr}}_{r}P\left[\exp\left(\oint_{C}A\right)\right]$ (IV.43)

Here $r$ denotes the representation. We will see in a second that the charged
line operator that we want comes from the fundamental representation $r=f$.

The 2d topological surface operator is called a Gukov-Witten surface operator
$G_{g}({{\mathcal{M}}}_{2})$ Gukov:2006jk ; Gukov:2008sn  and in this case it
is labeled by an element of $g\in\mathbb{Z}_{N}$. We will first study a
perhaps slightly heuristic explanation of its existence in the continuum: a
more convincing argument will require us to study a UV regularization on the
lattice.

The heuristic idea is the following: $G_{g}({{\mathcal{M}}}_{2})$ is defined
by saying that when it is inserted into the path integral, the boundary
conditions on the gauge field are modified so that holonomy of the gauge field
on an infinitesimal loop $C$ that winds around the codimension-2 surface
${{\mathcal{M}}}_{2}$ is in the same conjugacy class as a center element $g$,
i.e.

$P\exp\left(\oint_{C}A\right)=g$ (IV.44)

(One can also define these by the conjugacy classes of other group elements
that are not in the center; in this case they are however generically not
topological objects).

Physically, these objects can be thought of as inserting vortices which induce
a kind of holonomy for the gauge charged objects that move around them. This
holonomy acts as a phase on the fundamental Wilson line, i.e. if
$g=e^{\frac{2\pi ik}{N}}$ then we have Heidenreich:2021xpr :

$G_{g}(S^{2})W_{f}(C)=\frac{\mathop{\rm Tr}_{f}(g)}{\mathop{\rm
Tr}_{f}({\mathbf{1}})}W_{f}(C)=e^{\frac{2\pi ik}{N}}W_{f}(C)$ (IV.45)

and thus we see that it generates the desired linking equation (III.7) when a
Wilson line is inserted into the path integral.

(To understand the second equality, note that the holonomy constraint above
simply always contributes an extra constant factor of $g$ inside the path-
ordered exponential; as $g$ commutes which everything it is proportional to
the identity inside the trace and contributes at most a phase, and we can
extract that phase by writing it as the ratio of traces).

However we are looking at pure Yang-Mills theory, where there are no dynamical
degrees of freedom transforming in the fundamental. Do any dynamical degrees
of freedom see this surface operator? The dynamical degrees of freedom are the
gluons etc. which are in the adjoint. The holonomy picked up by an adjoint-
valued degree of freedom as it traverses the Gukov-Witten operator is given by
the Wilson line in the adjoint representation:

$G_{g}(S^{2})W_{\mathrm{Adj}}(C)=\frac{\mathop{\rm
Tr}_{\mathrm{Adj}}(g)}{\mathop{\rm
Tr}_{\mathrm{Adj}}({\mathbf{1}})}W_{\mathrm{Adj}}(C)=W_{\mathrm{Adj}}(C)$
(IV.46)

However, as the adjoint representation acts as conjugation $M\to
U^{\dagger}MU$, the center doesn’t act at all on the adjoint representation,
and the trace of any central element in the adjoint is the same as the trace
of the identity, $\mathop{\rm Tr}_{\mathrm{Adj}}(g)=\mathop{\rm
Tr}_{\mathrm{Adj}}(\mathbf{1})$. Thus none of the gluons can see the Gukov-
Witten surface operator, and it is topological.

In the above discussion one may worry that some subtleties are lost in the
continuum; we will thus now present the same argument with a lattice
description of the gauge theory. This has the benefit of making everything
extremely transparent, and we will also see how we can identify discrete
symmetries in a lattice model.

####  IV.4.2 SU(N) gauge theory on the lattice

In this section we will formulate pure $SU(N)$ gauge theory on the lattice.
This is a standard subject, and our discussion follows that of tong2018gauge .
Consider a hypercubic lattice $\mathbb{Z}^{4}$. The basic degree of freedom in
lattice gauge theory is a group element $U_{l}\in SU(N)$ defined on each of
the directed links $l$ of the lattice. We will sometimes use a notation
$l=\langle ij\rangle$ to denote a link from site $i$ to site $j$. Reversing
the orientation of the link corresponds to taking the inverse of the group
element, i.e.

$U_{\langle ij\rangle}=U^{\dagger}_{\langle ji\rangle}$ (IV.47)

Gauge transformations $\Lambda_{i}$ are $SU(N)$ valued functions defined on
the sites of the lattice, and they act on the link variables as

$U_{\langle ij\rangle}\to\Lambda_{i}U_{\langle
ij\rangle}\Lambda_{j}^{\dagger}$ (IV.48)

Morally speaking, to make the connection to the continuum gauge potential $A$,
we should view $U$ as an infinitesimal open Wilson line from $i$ to $j$:

$U_{\langle ij\rangle}=\exp\left(i\int_{i}^{j}A\right)\ .$ (IV.49)

The analogue of the 2-form field strength is defined on a plaquette, and is
constructed by taking the oriented trace of the four links on the plaquette:

$W_{\Box}=\mathrm{Tr}_{f}\prod_{l\in\Box}U_{l}$ (IV.50)

Unpacking this for a plaquette in the $(x,y)$ plane with bottom left corner at
$(x,y)$ we have:

$W_{\Box}=\mathrm{Tr}_{f}\left(U_{\langle(x,y),(x+1,y)\rangle}U_{\langle(x+1,y),(x+1,y+1)\rangle}U_{\langle(x+1,y+1),(x,y+1)\rangle}U_{\langle{(x,y+1),(x,y)}\rangle}\right)$
(IV.51)

Note that this is invariant under (IV.48) because it forms a tiny closed loop
and all of the $\Lambda_{i}$’s cancel.

We are now ready to form an action. The Wilson action for pure Yang-Mills is
the following sum over plaquettes:

$S[U]=-\frac{1}{g^{2}}\sum_{\Box}\left(W_{\Box}+W_{\Box}^{\dagger}\right)$
(IV.52)

Here $g^{2}$ is the Yang-Mills coupling.101010This discussion may make it seem
that a naive and generic discretization preserves all the global symmetries of
the continuum action. This is not true – the “magnetic” symmetry can be broken
by the existence of dynamical defects at the lattice scale Polchinski:2003bq
– and this is at the heart of phenomena such as Polyakov’s description of
confinement in (2+1)d $U(1)$ gauge theory on the latttice Polyakov:1975rs  and
the Berezinskii-Kosterlitz-Thouless transition in 2d
berezinskii1971destruction ; kosterlitz2018ordering . Recent work has shown
how these symmetries can be preserved on the lattice, see. e.g.
Sulejmanpasic:2019ytl ..

Exercise: By assuming that each link has length $a$ and appropriately Taylor
expanding the continuum field $A_{\mu}(x)$ defined in (IV.47), show that to
leading order in derivatives
$W_{\Box}=-\frac{a^{4}}{2}\mathrm{Tr}_{f}F_{\mu\nu}F_{\mu\nu}+\cdots$ (IV.53)
where $F$ is the usual non-Abelian field strength and there is no sum over
indices on the left-hand side (rather the $\mu\nu$ indicate which plaquette we
are discussing). Thus show that the lattice Wilson action reduces to the
regular continuum action (IV.40) to leading order in derivatives.

The partition function is then given by integrating over all the group
elements on each link:

$Z=\int[dU_{l}]\exp(-S[U])$ (IV.54)

There exists a well-defined invariant integration measure on group elements
called the Haar measure, which basically does exactly what you expect it to do
and is invariant under the group action. The partition function is a
discretely infinite number of integrals over $U$ and can be (fruitfully) done
on a computer.

Now, let us understand the global symmetries of the theory defined by (IV.54).
The fundamental Wilson line is defined in the obvious way, by taking products
of $U$ along a closed path $C$:

$W_{r}(C)=\mathrm{Tr}_{r}\left(\prod_{l\in C}U_{l}\right)$ (IV.55)

Let us now turn to the Gukov-Witten surface operator.111111I am thankful to T.
Sulejmanpasic for discussions on the content of this section.

To begin, note that the Wilson action that we defined above has a natural
“slot” to which we can couple a $\mathbb{Z}_{N}$-valued 2-form source.
Consider defining a $\mathbb{Z}_{N}$ valued function on the plaquettes of the
lattice by $b_{\Box}\in\mathbb{Z}_{N}$. Then we can construct a deformed
version of $W_{\Box}$ as

$W_{\Box}(b_{\Box})=\mathrm{Tr}_{f}\left(b_{\Box}\prod_{l\in\Box}U_{l}\right)\
.$ (IV.56)

i.e. we are twisting the product of $U$’s by the appropriate $b$ on the
plaquette. It doesn’t matter where we place the $b$ in the string of $4$
$U$’s, as it commutes with everything. The action and partition function may
be defined in the presence of this source.

$S[U;b]=-\frac{1}{g^{2}}\sum_{\Box}\left(W_{\Box}(b_{\Box})+W_{\Box}^{\dagger}(b_{\Box})\right)\qquad
Z[b]=\int[dU_{l}]\exp(-S[U;b])$ (IV.57)

Now to define the Gukov-Witten operator $G_{g}({{\mathcal{M}}}_{2})$: let
${{\mathcal{M}}}_{2}$ be a 2d surface on the dual-lattice121212The dual
lattice has a precise definition, but for the cubic lattice its the set of all
points that are “halfway between” the points of the lattice, i.e. $({1\over
2},{1\over 2},{1\over 2},{1\over 2})$ \+ (integer offsets). A codimension-2
surface naturally intersects a set of plaquettes on the original lattice (as
shown in a 3d example in Figure 11).

Figure 11: A codimension-$2$ surface on the dual lattice intersects plaquettes
of the original lattice, shown here in a 3d example; red line represents
codimension-$2$ surface, and shaded red plaquettes are intersected by it.

For each plaquette that intersects ${{\mathcal{M}}}_{2}$, set $b_{\Box}=g$.
(For all other plaquettes, set $b_{\Box}=\mathbf{1}$). The resulting choice of
$b_{{{\mathcal{M}}}_{2}}$ can be thought of as defining a new partition
function which depends on a choice of surface corresponding to the insertion
of the operator131313For an interesting study in 2d of related operators see
Nguyen:2021naa ..

$Z[b_{{{\mathcal{M}}}_{2}}]=\langle G_{g}({{\mathcal{M}}}_{2})\cdots\rangle$
(IV.58)

To see that this is topological, let us consider trying to deform the surface.
Most of the action is in two directions transverse to the defect, so let us
focus on those, as in Figure 12. Note that if we do the following change of
variables in the picture:

$U_{4}\to g^{-1}U_{4}$ (IV.59)

it has the effect of setting $b_{\Box_{1}}=\mathbf{1}$ but setting
$b_{\Box_{2}}=g$; thus it has moved the surface operator over by one site, but
leaves the partition function invariant (as it is just a change of variables
in the path integral). By performing operations of this form one can deform
the operator however we like; thus there exists a topological operator living
on 2d surfaces in the dual lattice. Zooming out to long distances the
distinction between dual and original lattices goes away and this becomes a
smoothly deformable continuum operator.

Figure 12: Moving the Gukov-Witten surface operator around by a change of
variables in the lattice path integral.

Exercise: Prove that the linking of the lattice Gukov-Witten surface operator
$G_{g}({{\mathcal{M}}}_{2})$ with the fundamental Wilson line (IV.55)
satisfies the desired linking equation (IV.45).

The existence of a topological operator $G_{g}({{\mathcal{M}}}_{2})$ is a
statement that can easily be lifted to the continuum. In fact this is a
consequence of a larger invariance of the partition function in the presence
of the source $b$. Consider a $\mathbb{Z}_{N}$ valued function $g_{l}$ on the
links $l$. Assume that this acts on the dynamical link variables $U_{l}$ and
the plaquette variables $b_{\Box}$ as follows:

$U_{l}\to U_{l}^{\prime}=U_{l}g_{l}\qquad b_{\Box}\to
b_{\Box}^{\prime}=b_{\Box}\prod_{l\in\Box}g_{l}^{\dagger}$ (IV.60)

(where the latter product is again taken in an oriented fashion). This leaves
the twisted plaquette product $W_{\Box}$ – and thus the whole action $S[U;b]$
– invariant. The partition function is thus invariant under

$Z[b^{\prime}]=Z[b]$ (IV.61)

This invariance is equivalent to the existence of the $\mathbb{Z}_{N}$ 1-form
global symmetry. Note that it is a discrete version of the sort of invariance
we saw for the $U(1)$ Maxwell theory around (IV.9). Importantly, it not the
same as the $SU(N)$ gauge redundancy discussed around (IV.48), which involved
an $SU(N)$ group element on each lattice site, not a $\mathbb{Z}_{N}$ on the
links. We will shortly see that one can preserve the gauge redundancy while
breaking the 1-form global symmetry.

####  IV.4.3 Confinement and the 1-form symmetry

Having now carefully understood that pure $SU(N)$ gauge theory has a
$\mathbb{Z}_{N}$ symmetry under which the fundamental Wilson loop is charged,
we can ask about the different phases:

  1. 1.

Confined phase: here the 1-form symmetry is unbroken, and the Wilson loop
behaves as an area law: $\langle W_{F}(C)\rangle\sim\exp(-T\mbox{Area}(C))$.
The physics interpretation is that the Wilson loop creates a non-Abelian flux
tube which has a finite tension $T$; it is these flux tubes which the 1-form
symmetry counts (mod N). This is the phase where we normally end up for non-
Abelian gauge theory.

  2. 2.

Deconfined phase: here the 1-form symmetry is spontaneously broken, and the
Wilson loop behaves as a perimeter law $\langle
W_{F}(C)\rangle\sim\exp(-TL[C])$. The flux tubes have melted and condensed.

Note that from this point of view the confinement/deconfinement transition is
philosophically the same as the phase transitions that we know from the
ordinary Landau paradigm, just involving a 1-form symmetry rather than a
0-form one!

We can now ask the following question: what happens if we add matter to the
theory, e.g. if we add a scalar charged in the fundamental under $SU(N)$:

$S[A,\phi]=\int d^{4}x\left(\frac{1}{2g^{2}}\mathop{\rm
Tr}(F_{\mu\nu}F^{\mu\nu})+(D\phi)^{\dagger}D\phi+m^{2}\phi^{\dagger}\phi\right)$
(IV.62)

where $D_{\mu}\phi=\partial_{\mu}\phi+iA_{\mu}^{a}t^{a}\phi$ with $t^{a}$ the
fundamental representation matrices.

In this case the 1-form symmetry is explicitly broken, i.e. the putative
Gukov-Witten surface operator $G_{g}({{\mathcal{M}}}_{2})$ is no longer
topological.

We can see this in the continuum by noting that the quanta of the $\phi$ field
trace out effective worldlines in spacetime, where the coupling of each
worldline to the gauge field is given by a fundamental Wilson loop $\langle
W_{f}(C)\rangle$; thus $G_{g}({{\mathcal{M}}}_{2})$ now feels a phase from the
dynamical degrees of freedom, spoiling its topological-ness.

More concretely, we can also see this on the lattice. To add fundamental
matter to the Wilson lattice action, we consider a field
$\phi_{i}\in\mathbb{C}^{N}$ living on each site $i$ of the lattice. Under the
gauge transformation (IV.48) we take $\phi_{i}$ and $U_{\langle ij\rangle}$ to
transform as

$U_{\langle ij\rangle}\to\Lambda_{i}U_{\langle
ij\rangle}\Lambda_{j}^{\dagger}\qquad\phi_{i}\to\Lambda_{i}\phi_{i}$ (IV.63)

so that the coupling $\phi_{i}^{\dagger}U_{\langle ij\rangle}\phi_{j}$ is a
gauge-invariant lattice kinetic term for $\phi$. We can now consider the
Wilson action with fundamental matter:

$S[U,\phi]=-\frac{1}{g^{2}}\sum_{\Box}\left(W_{\Box}+W_{\Box}^{\dagger}\right)+t\sum_{\langle
ij\rangle}\phi_{i}^{\dagger}U_{\langle ij\rangle}\phi_{j}$ (IV.64)

(We can also add a further on-site mass term which we neglect for simplicity).
If we now try to construct the topological Gukov-Witten operator through the
arguments above, we will see that it is simply not possible; the existence of
the $\phi$’s renders impossible the manipulations in (IV.59).

This means that once fundamental charged matter exists, there is no longer a
clean symmetry-based distinction between the confined and deconfined phases;
in all phases we will find that the Wilson law exhibits a perimeter law.
(Physically, if you try to create an extended flux tube, it is now allowed to
break into two pieces with a $\phi$ particle ending each tube). Thus there is
no order parameter for confinement in this case, because there is no global
symmetry. (Note that QCD also has matter in the fundamental representation,
and thus does not permit a simple order parameter for confinement.).

###  IV.5 Gauge theory: $\mathbb{Z}_{2}$

A careful reader will have noticed above that most of the structure of the
non-Abelian gauge symmetry played essentially no role in this discussion. One
might then reasonably ask: what is the simplest theory that has the structure
above, i.e. that of a discrete 1-form symmetry which could be spontaneously
broken?

The simplest is the $\mathbb{Z}_{2}$ lattice gauge theory. This is essentially
the same as above, except that instead of having a $SU(N)$ element on each
link we have a $\mathbb{Z}_{2}$ element, i.e. a single sign which can be
either positive or negative:

$a_{l}=\pm 1$ (IV.65)

The analogue of the field strength is also valued in $\pm 1$:

$W_{\Box}=\prod_{l\in\Box}a_{l}$ (IV.66)

The action is simply

$S[a]=-\beta\sum_{\Box}W_{\Box}\qquad Z=\sum_{\\{a_{l}\\}}\exp(-S[a])$ (IV.67)

There is a gauge redundancy given by

$a_{\langle ij\rangle}\to a_{\langle ij\rangle}^{\prime}=\Lambda_{i}a_{\langle
ij\rangle}\Lambda_{j}$ (IV.68)

where $\Lambda_{i}$ is a $\mathbb{Z}_{2}$-valued field defined on site.

Importantly, this theory also has a $\mathbb{Z}_{2}$ 1-form global symmetry.
The charged line operator is the Wegner-Wilson line operator, given by by
$W[C]=\prod_{l\in C}a_{l}$. The co-dimension $2$ charge defect – i.e. the
analogue of the Gukov-Witten surface operator – can be constructed by an
argument similar to that above, and turns out to take the explicit form:

$G(S)=\prod_{\Box\in S}\exp(-2\beta W_{\Box})$ (IV.69)

where again the product over plaquettes is taken over those that intersect the
surface in the dual lattice (here the factor of 2 comes from the fact that we
are changing the sign of the appropriate plaquette term in the action).

This theory has a confined phase (unbroken 1-form symmetry) at $\beta\approx
0$ and a deconfined phase (spontaneously broken 1-form symmetry) at $\beta\gg
1$, where the precise location of the critical point can easily be found by
numerical simulation. Much more can be said about this, and we will just make
a few more quick remarks.

####  IV.5.1 Kramers-Wannier duality and the Ising spin model

In 3d, this theory is dual by Kramers-Wannier duality Wegner1971  – i.e. a
non-local rewriting of the theory – to the regular 3d Ising spin model, which
has action

$S=-\tilde{\beta}\sum_{\langle ij\rangle}\sigma_{i}\sigma_{j}$ (IV.70)

where $\sigma_{i}=\pm 1$, and where we have $\tilde{\beta}=-{1\over
2}\log(\tanh\beta)$.

Physically speaking, this mapping takes the worldsheets of 2d flux tubes in
the gauge model to domain walls that separate up from down spins in the spin
model.

The global symmetries may be confusing here, as clearly the model (IV.70) has
a $0$-form $\mathbb{Z}_{2}$ global symmetry $\sigma_{i}\to-\sigma_{i}$, and
not a 1-form global symmetry. The precise statement of Kramers-Wannier duality
for the 3d Ising model is that the $\mathbb{Z}_{2}$-invariant sectors are
mapped into each other, and a genuine charged operator on one side is mapped
into a non-genuine charged operator (i.e. something with a tail attached) on
the other.

Of course, this is a very well-studied model for magnetism. The action
penalizes configurations where $\sigma_{i}$ differs from its neighbours, and
thus the low-temperature phase $(\tilde{\beta}\to\infty)$ has all of the
$\sigma_{i}$ with the same sign, and the $\mathbb{Z}_{2}$ spin symmetry
spontaneously broken, the ferromagnetic phase. As we increase temperature
(i.e. take $\tilde{\beta}\to 0$) we eventually hit a phase transition at
$\tilde{\beta}\approx 0.221$141414See Xu:2018hwn  for a recent measurement of
this critical coupling from Monte Carlo simulations. and the model enters a
disordered paramagnetic phase. The continuous critical point separating these
two – the 3d Ising transition – is a strongly coupled fixed point about which
we nevertheless know a lot. Currently the most precise estimates of the
critical exponents come from the conformal bootstrap El-Showk:2012cjh  (see
Poland:2018epd  for a review).

####  IV.5.2 Higgs/confinement continuity

Let us study slightly further this model in 3d. We can imagine coupling in
$\mathbb{Z}_{2}$ matter to this gauge field, i.e. introduce $\phi_{i}=\pm 1$,
a $\mathbb{Z}_{2}$-valued field on the sites who transforms under the gauge
transformations as

$\phi_{i}\to\Lambda_{i}\phi_{i}$ (IV.71)

We then have the following model, the $\mathbb{Z}_{2}$ analogue of (IV.64):

$S[a,\phi]=-\beta\sum_{\Box}W_{\Box}-J\sum_{\langle
ij\rangle}\phi_{i}a_{\langle ij\rangle}\phi_{j}$ (IV.72)

This model illustrates some important physical points:

  1. 1.

First, by arguments identical to those around (IV.64), the presence of matter
breaks the $\mathbb{Z}_{2}$ 1-form symmetry; thus at generic points in the
$(\beta,J)$ space the model has no global symmetries.

  2. 2.

Here $J$ controls the matter fluctuations and $\beta$ the gauge fluctuations.
What is the phase diagram as a function of $J$ and $\beta$? If we set $J\to 0$
then there is a deconfined phase at larger $\beta$ and a confined phase at
small $\beta$.

Let’s now consider the limit where $\beta\to\infty$; this essentially
suppresses all the gauge field fluctuations, so we expect to find a setup
where $W_{\Box}=1$ on all plaquettes, which means that $a_{\langle
ij\rangle}=1$ (or its gauge-equivalent) on all links. We thus end up with the
ordinary Ising spin model (IV.70) expressed in terms of the $\phi_{i}$; as
described above this model has a phase transition at some critical point
$J^{*}$.

Note that this phase transition is – morally – condensing the $\phi_{i}$
fields. So in conventional language, we would call it the “Higgs” phase. We
say morally because the $\phi_{i}$ are gauge charged and thus we cannot really
speak about a condensate of them, but it is a fact that the phase transition
happens.

Figure 13: Cartoon phase diagram of model described by (IV.72); note that one
can move continuously from the Higgs to the confined phase. Details of the
physics along the diagonal have been omitted, see Somoza:2020jkq  for a
discussion.

  3. 3.

The phase diagram of this model, as shown in Figure 13 is now somewhat
interesting. Note that there are basically two phases; one that is
continuously connected to the deconfined phase of the gauge theory – we will
call this the topological phase – and another, which is continuously connected
both to the Higgs phase on one side, and the confined phase on the other. If
nothing else happens, this implies the baffling statement that the Higgs and
confined phases are continuously connected PhysRevD.19.3682 .

From the point of view of gauge theory as formulated as a perturbation theory
in the continuum, this statement is very surprising. However it is indeed
true; the point is simply that there are no global symmetries that can be used
to diagnose the phases, and thus, there is nothing stopping the Higgs and
confined phases from being connected to each other. The status of
Higgs/confinement continuity in more complicated theories – such as, e.g. QCD
– is a matter of active study (see e.g. Dumitrescu:2023hbe ; Cherman:2024exo
).

  4. 4.

It is not an accident that the phase diagram appears symmetric with respect to
reflection about the diagonal. There is in fact a self-duality symmetry that
switches the two sides; see Somoza:2020jkq  for a discussion of the rich
physics appearing at the phase transition at the self-dual point. As of this
writing, there is still not yet a useful continuum description for this phase
transition, which is possibly the simplest beyond-Landau critical point.

###  IV.6 Effective description of spontaneously broken $\mathbb{Z}_{N}$
1-form symmetry: topological order

We now turn to a different question: suppose we have a theory that has a
spontaneously broken $\mathbb{Z}_{N}$ 1-form symmetry, such as deconfined
$SU(N)$ gauge theory, or maybe the deconfined phase of the $\mathbb{Z}_{2}$
gauge theory described above. What is the effective description of the long
distance physics? (Here we will be motivated by $SU(N)$ gauge theory and its
associated $\mathbb{Z}_{N}$ 1-form symmetry).

We should first ask: what are the long-distance degrees of freedom?
Generically, there will be a gap to all local degrees of freedom, but we
should be able to access objects corresponding to the fundamental Wilson line
and the Gukov-Witten surface operator, which have a non-trivial braiding phase
when they link as in (IV.45)

$G(S^{2})W(C)=e^{\frac{2\pi i}{N}}W(C)$ (IV.73)

where here we have omitted the subscripts and are now referring to the minimal
charge object of each class. Note that though $G(S^{2})$ was topological by
construction, in the extreme infrared limit $W(C)$ is also topological – this
is because it has vanishing correlation with all other operators (which
themselves have only exponentially suppressed correlations with each other),
but it is nevertheless not zero (as it would have been if it had an area law).

####  IV.6.1 BF theory

We thus seek a topological field theory to capture the physics of this
braiding. Our discussion will be in 4d with an eye towards the $SU(N)$ gauge
theory, but by modifying a few indices here and there it works for
$\mathbb{Z}_{N}$ p-form symmetry in any number of dimensions.

The required object is called BF theory, and can be defined in terms of a
local action involving a 1-form and 2-form gauge field $A_{1}$ and $B_{2}$
respectively:

$S[A,B]=\frac{iN}{2\pi}\int_{{{\mathcal{M}}}_{4}}B\wedge
dA=\frac{iN}{2\pi}\int_{{{\mathcal{M}}}_{4}}A\wedge dB$ (IV.74)

This is a topological field theory with a long history horowitz1989exactly ;
see Banks:2010zn ; Maldacena:2001ss  for discussions in the high-energy
literature. It is a kind of mixed, higher-form Chern-Simons theory. The metric
does not appear in its construction: it does not care about distances, and
cares only about topology. As we will see, it can also be thought of as a
gauge theory with a $\mathbb{Z}_{N}$ gauge group.

This action is invariant under the following gauge redundancies:

$B_{2}\to B_{2}+d\Lambda_{1}\qquad A_{1}\to A_{1}+d{\lambda}_{0}\ .$ (IV.75)

It’s important that both of these are compact gauge fields, i.e. the gauge
transformations are valued in $U(1)$ and just as around (IV.11) the associated
monopole numbers are quantized appropriately, i.e. if we have $F=dA$ and
$H=dB$ then over all 2 and 3-cycles respectively we have:

$\oint_{{{\mathcal{M}}}_{2}}F_{2}=2\pi\mathbb{Z}\qquad\oint_{{{\mathcal{M}}}_{3}}H_{3}=2\pi\mathbb{Z}$
(IV.76)

This imposes a restriction on the parameter $N$. Consider studying the action
the theory on $S^{1}\times S^{3}$. Now consider doing a large gauge
transformation by gauge parameter $e^{i\lambda}$, where $\lambda$ winds once
around the $S^{1}$, i.e. if the $S^{1}$ coordinate is $\phi\in[0,2\pi]$

$\lambda=\phi\ .$ (IV.77)

Here the word “large” refers to the fact that this gauge transformation cannot
be continuously deformed to the identity.151515Note that in the literature a
“large gauge transformation” can also sometimes refer to a gauge
transformation on a non-compact space, where the gauge parameter does not die
away at infinity. This is a different situation, as those large gauge
transformations typically can be continuously connected to the identity. They
are also (usually) actually not “gauge”, in they generate global symmetries,
i.e. actual physical invariances of the system, unlike the large gauge
transformations discussed above, which are redundancies of the description.

The gauge transformation $e^{i\lambda}$ is single-valued on the circle, and
thus this is a perfectly valid gauge transformation to do. Under this
transformation $A\to A^{\prime}=A+d\lambda=A+d\phi$

$S[A^{\prime},B]-S[A,B]=\frac{iN}{2\pi}\int_{S^{1}\times S^{3}}d\phi\wedge
H=\frac{iN}{2\pi}\left(\int_{S^{1}}d\phi\right)\left(\int_{S^{3}}H\right)=2\pi
iNq$ (IV.78)

where $q$ is the integer-quantized “monopole number” of the integral of $H$
over $S^{3}$. Note the horrifying fact that the action is not invariant under
this large gauge transformation. We are saved by the fact that we do not study
the action itself; rather this is an intrinsically quantum theory which makes
sense only when we insert the action into a path integral as

$Z=\int[dA][dB]\exp(-S[A,B])$ (IV.79)

and thus what we actually need is for the exponential of $S$ to be invariant.
This will be the case provided we have

$N\in\mathbb{Z}\ .$ (IV.80)

meaning that the shift in $S$ affects the partition function by an invisible
factor of $e^{2\pi i}$.

We thus see that the theory defined by (IV.74) has its parameter $N$
quantized. (An alternative classic argument involving defining Chern-Simons
theory as the boundary of a higher-dimensional manifold can be found in
Witten:2003ya ). This rigidity of the theory is generally the case for
topological field theories, which are discrete objects that do not permit
continuous deformations.

We now study the dynamics. The local equations of motion are

$dA=0\qquad dB=0$ (IV.81)

and thus there are no local degrees of freedom. However there are extended
operators we can insert on closed 1-cycles $C_{1}$ and 2-cycles
${{\mathcal{M}}}_{2}$ respectively:

$W(C_{1})=\exp\left(i\oint_{C_{1}}A\right)\qquad
G({{\mathcal{M}}}_{2})=\exp\left(i\oint_{{{\mathcal{M}}}_{2}}B\right)$ (IV.82)

These objects are invariant under the gauge transformations (IV.75). We now
make the main dynamical statement – in BF theory these two objects have a
nontrivial $\mathbb{Z}_{N}$ braiding, i.e.

$\big{\langle}W(C_{1})G({{\mathcal{M}}}_{2})\big{\rangle}\sim\exp\left(\frac{2\pi
i}{N}L(C_{1},{{\mathcal{M}}}_{2})\right)$ (IV.83)

where $L$ is the linking number of $C_{1}$ and ${{\mathcal{M}}}_{2}$, a well-
defined object in four dimensions.

The proof of this is not completely obvious, and is a good exercise.

Exercise: Prove the linking relation (IV.83). Hint: consider the equations of
motion with the surface operators inserted and study the on-shell action for a
simple configuration of $C_{1}$ and ${{\mathcal{M}}}_{2}$.

A solution to the exercise is provided below. Insert the two surface operators
into action to find:

$S[A,B]=\frac{iN}{2\pi}\int B\wedge
dA+i\oint_{C_{1}}A+i\oint_{{{\mathcal{M}}}_{2}}B$ (IV.84)

Varying with respect to $A$ and $B$ we find that the equations of motion
become

$\displaystyle\frac{N}{2\pi}dA(x)$
$\displaystyle=-\delta_{{{\mathcal{M}}}_{2}}(x)$ (IV.85)
$\displaystyle\frac{N}{2\pi}dB(x)$ $\displaystyle=-\delta_{C_{1}}(x)\ .$
(IV.86)

Let us denote the solutions to this set of equations to be $\overline{A}$,
$\overline{B}$. Plugging this in to find the on-shell action we find that two
terms cancel and we are left with:

$S[\overline{A},\overline{B}]=i\oint_{C_{1}}\overline{A}$ (IV.87)

In other words, we should determine the $\overline{A}$ sourced by the
insertion of $G({{\mathcal{M}}}_{2})$ through (IV.85) and evaluate it on the
location of the insertion of $W(C_{1})$.

Let us now assume that $C_{1}$ links ${{\mathcal{M}}}_{2}$ once. The geometry
is the same as in Figure 10. Consider integrating (IV.85) over a small ball
$B_{2}$ which is bounded by $C_{1}$; this intersects ${{\mathcal{M}}}_{2}$ at
a point and we thus find that

$\frac{N}{2\pi}\int_{B^{2}}d\overline{A}=\frac{N}{2\pi}\oint_{C_{1}}\overline{A}=-1$
(IV.88)

Plugging everything in, we find that in the case where the two manifolds link
once we have:

$S[\overline{A},\overline{B}]=-\frac{2\pi i}{N}$ (IV.89)

From this it is clear that if they had linked multiple times we would obtain
an extra factor on the right-hand side. On the other hand, if they don’t link
at all we obtain zero; we have thus found the relation (IV.83) (up to an
overall sign which depends on a choice of orientation).

Thus we see that we have realized the desired linking algebra (IV.73). A phase
described by such a topological field theory is usually called an example of
topological order, and is often exhibited as an example of a beyond-Landau
phase of matter wen1990topological ; wen2004quantum . This name predates the
technology of higher-form symmetry: in this section we have just shown that
the spontaneous breaking of a 1-form symmetry is an example of topological
order, and thus can be understood from the point of view of an enlarged Landau
paradigm involving higher-form symmetry.

####  IV.6.2 Ground states in a topologically ordered phase

If we spontaneously break a conventional 0-form $\mathbb{Z}_{N}$ symmetry, we
have $N$-fold degenerate ground states which are acted upon by the action of
the symmetry. They are precisely degenerate in the limit of infinite volume.

Here we will discuss the higher-form analogue of this phenomena; again if we
break a $1$-form $\mathbb{Z}_{N}$ symmetry, we will have degenerate ground
states. In this case their number is determined by the topology of the spatial
manifold.

As an example, let us consider quantizing the theory on $\mathbb{R}\times
S^{2}\times S^{1}$, where the first $\mathbb{R}$ is “time”. (Note that this is
the first time in these notes that we have worked in a Hamiltonian framework).
It is instructive to quantize the theory carefully. We just note the following
points, whose verification is left as an exercise:

  1. 1.

There are no local degrees of freedom. There may still be “delocalized” states
that live on the whole spatial manifold.

  2. 2.

The Hamiltonian is zero, so all of these states must have vanishing energy:
they are all “ground states”.

  3. 3.

The only non-trivial operators that act on the space of states are the
following

$W\equiv\exp\left(i\oint_{S^{2}}B\right)\qquad
G\equiv\exp\left(i\oint_{S^{1}}A\right)$ (IV.90)

These are the only topologically non-trivial operators that we can construct.
As the equations of motion set $dA=dB=0$ any topologically trivial operators
can be shrunk to zero and collapsed, and have no action on any state.

Note also that these are the objects that we saw had a non-trivial braiding in
Euclidean space (IV.83). Here we are interpreting them as quantum operators
acting on the Hilbert space, i.e. they act at a given time. One can directly
show from the canonical commutation relation of the local fields $A$ and $B$
that the non-local operators satisfy the following equal-time commutation
relation:

$GW=e^{\frac{2\pi i}{N}}WG,$ (IV.91)

We now seek to find the space of states, i.e. we seek to find a representation
of the algebra (IV.91). There is no one-dimensional representation, and thus
the ground state is necessarily not unique. The smallest representation is
$N$-dimensional, with basis elements which we label by
$|n\rangle,n\in\\{0,1,\cdots(N-1)\\}$. The operators act as

$W|n\rangle=|(n+1)\;\mbox{mod }N\rangle\qquad G|n\rangle=\exp\left(\frac{2\pi
in}{N}\right)|n\rangle$ (IV.92)

These $N$ states are the ground states of BF theory on $S^{2}\times S^{1}$.
Physically, the state $|n\rangle$ corresponds to having $n$ flux tubes
threading the $S^{1}$; acting with the Wilson line on the $S^{1}$ creates
another flux tube, and $G$ measures the number of tubes (mod $N$).

Note that this state exists only because of the non-trivial topology; had we
played the same game on $S^{3}$ we would have found a single unique ground
state. This is a hallmark of topological order.

Exercise: Prove the commutation relation (IV.91).

Exercise: Consider a situation where there are dynamical string worldsheets –
with tension $T$ – that couple to the 2-form $B_{2}$. Compute in some suitable
approximation the splitting of the degenerate energy levels we have found
above. Hint: – you should find $\Delta E\sim\exp(-4\pi TR^{2})$, where the
$S^{2}$ has radius $R$.

There is much more to say about topological order. We will make only a few
more remarks:

  1. 1.

A simple and extremely instructive Hilbert space realization of the
topological phase of the $\mathbb{Z}_{2}$ gauge theory in $(2+1)d$ is given by
the so-called toric code Kitaev:1997wr .

  2. 2.

These lectures were first given at the Jena school for theoretical physics,
where the other lecture series was about entanglement entropy. In fact, a
system that has topological order will exhibit a signature of it in its
entanglement entropy. There is a reduction of entropy arising from the fact
that every one of the un-endable-strings that we discuss that enters a region
of space must also exit the same region of space. In the two-dimensional toric
code case discussed above this so called “topological entanglement entropy”
Kitaev:2005dm ; PhysRevLett.96.110405  is a constant contribution which takes
the form

$S(\mbox{disc})=\mbox{non-universal area law}-\log 2$ (IV.93)

Clearly for a $\mathbb{Z}_{N}$ gauge theory the $2$ is replaced by an $N$. The
higher dimensional case is discussed from the microscopic theory in
Grover:2011fa  and from the effective BF description in Fliss:2023uiv 161616As
of this writing there appears to be a slight discrepancy between these two
descriptions.. As described extensively in Kitaev:2005dm ;
PhysRevLett.96.110405 , the $\log N$ contribution may be cleanly separated
from the area law by adding and subtracting the entanglement entropy defined
on different spatial regions.

  3. 3.

The usual first example of topological order in most textbooks is actually
usually not the $SU(N)$ gauge theory, but rather the fractional quantum Hall
effect. It turns out that if you confine many electrons to a plane and apply a
strong magnetic field, the frustrated dynamics arising from the interaction of
Landau levels and Coulomb repulsion results in a topologically ordered state,
of which the simplest example be described by the following Chern-Simons
action:

$S[A]=\frac{ik}{8\pi}\int_{{{\mathcal{M}}}_{3}}A\wedge dA$ (IV.94)

where $A$ is not the photon of real life, but rather a new emergent gauge
field. There is again an emergent $\mathbb{Z}_{k}$ 1-form symmetry that is
spontaneously broken, but its realization is slightly different due to the
different structure of the Chern-Simons term. See Tong:2016kpv  for an
entrance to this fascinating field.

##  V Non-invertible symmetries: the axial anomaly

We will now discuss a different generalization of the idea of symmetry – non-
invertible symmetries.

Everything we have discussed so far has involved topological charge defects,
whose fusion obeys a group composition law (II.27):

$U_{g}({{\mathcal{M}}}_{d-1})U_{g^{\prime}}({{\mathcal{M}}}_{d-1})=U_{gg^{\prime}}({{\mathcal{M}}}_{d-1}),$
(V.1)

where $g,g^{\prime}\in G$ with $G$ some group. In particular, this means that
given a particular charge defect labeled by $g$, there is always an inverse
element $g^{-1}$ such that

$U_{g}({{\mathcal{M}}}_{d-1})U_{g^{-}1}({{\mathcal{M}}}_{d-1})=\mathbf{1}$
(V.2)

However one could ask – is it possible for there to exist topological defects
that obey a more general fusion rule, e.g. of the form?

$U_{a}({{\mathcal{M}}}_{d-1})U_{b}({{\mathcal{M}}}_{d-1})=\sum_{c}N^{c}_{ab}U_{c}({{\mathcal{M}}}_{d-1})$
(V.3)

where, e.g. the right hand side involves a non-trivial sum, rather than just a
single element? In that case it is possible that some elements will not have
an inverse. This is the idea of non-invertible symmetries.

Examples of this sort of phenomena have been known for a very long time in 2d
CFT verlinde1988fusion . Their study has exploded recently, with many new and
physically relevant examples found in higher dimensions. We will not make an
attempt to survey these developments here, but instead refer the reader to
recent reviews Cordova:2022ruw ; Schafer-Nameki:2023jdn ; Shao:2023gho  which
provide an entry point to the literature.

Before moving on to our main focus, we do however mention one important
example, which exists in the familiar 2d Ising model. This theory has an
ordinary 0-form invertible $\mathbb{Z}_{2}$ symmetry that acts on the spins as
$\sigma\to-\sigma$, and this symmetry is associated with an topological line
in the manner that is now familiar. However, when tuned to its critical point,
the model has an extra topological line which is non-invertible in the sense
described above. One way to understand this line is to consider taking half of
the 2d space and performing Kramers-Wannier duality on it. As this is a self-
duality for the 2d Ising model at the critical point, it will map the bulk of
the space to itself. However the junction between the original model and the
dual model has some non-trivial object living on it. This turns out to be a
topological line which is non-invertible in the sense described above
Frohlich:2004ef ; Chang:2018iay . It is beautifully reviewed from several
different points of view in Shao:2023gho .

In these notes we will now study how these symmetries present us with a new
way to think about the venerable Adler-Bell-Jackiw anomaly.

###  V.1 Different types of anomalies

Let us remind ourselves about the physics of anomalies, starting from the
example of a massless Dirac fermion in 4d. Consider first the theory of a
Dirac fermion alone, which we couple to an external $U(1)$ gauge field source
$a$:

$S[\psi;a]=\int d^{4}x{\bar{\psi}}\left(\not{\partial}-i\not{a}\right)\psi$
(V.4)

Let us remind ourselves of the symmetries. The action above is invariant under
two possible phase rotations of the Dirac fermion, $U(1)_{V}$ and $U(1)_{A}$:

$\psi\to e^{\frac{i{\alpha}}{2}}\psi\qquad\psi\to
e^{\frac{i{\alpha}}{2}{{\gamma}}^{5}}\psi$ (V.5)

which have corresponding currents

$j^{\mu}_{V}={\bar{\psi}}{{\gamma}}^{\mu}\psi\qquad
j^{\mu}_{A}={\bar{\psi}}{{\gamma}}^{5}{{\gamma}}^{\mu}\psi$ (V.6)

Above we have coupled an external gauge field source to the vector current.
Now it is well-known that if we insert this classical action into a path
integral as

$Z[a]=\int[d\psi d{\bar{\psi}}]\exp(-S[\psi;a])$ (V.7)

then in the quantum theory with the external source turned on the axial
current is no longer conserved Adler:1969gk ; Bell:1969ts :

$d\star j_{A}=\frac{1}{4\pi^{2}}f\wedge f\qquad f=da$ (V.8)

The result above is one-loop exact.

Note however that it is not conserved in a very specific way: the
nonconservation of the axial current is precisely determined by the external
source. This is called an ’t Hooft anomaly.

From the point of view of obtaining predictive power over the theory, a ’t
Hooft anomaly is generally better than not having any anomaly at all, as one
can obtain non-trivial constraints over the dynamics; after all, we can always
choose to set the applied source to zero (so that we have a conserved current
again), but we can do more. For example, it is clear that a theory with (V.8)
cannot have a trivial unique gapped ground state, as then one could imagine a
setup where the right-hand side of the equation (which we can control freely)
is not zero, but the left hand side (which is constrained by the gapped
dynamics) is.

However, in these lectures we will not be discussing ’t Hooft anomalies.
Instead we want to consider not the theory above – where $a$ was frozen – but
rather massless QED. To that end we will promote $a$ to a dynamical gauge
field $A$, with its own kinetic term:

$S[\psi,A]=\int
d^{4}x\left(\frac{1}{4e^{2}}F^{2}+{\bar{\psi}}\left(\not{\partial}-i\not{A}\right)\psi\right)\qquad
F=dA,$ (V.9)

and now also integrate over it in the path integral, i.e.

$Z=\int[dA][d\psi]\exp(-S[\psi,A])$ (V.10)

Let us now revisit the anomaly equation (V.8), which now has a very different
character:

$d\star j_{A}=\frac{1}{4\pi^{2}}F\wedge F$ (V.11)

as the right-hand side is now a dynamical operator, which fluctuates. Thus it
now naively seems like we have lost the symmetry entirely – the current is
just not conserved any more. We will refer to this kind of anomaly – where the
right-hand side is an operator – as an Adler-Bell-Jackiw anomaly171717This
nomenclature seems a little imperfect, but I believe it is now standard..

Let us nevertheless try to make sense of it. The first obvious thing to do is
consider defining the following shifted current:

$\star\tilde{j}_{A}=\star j_{A}-\frac{1}{4\pi^{2}}A\wedge F$ (V.12)

This satisfies $d\star\tilde{j}_{A}=0$. However it does not actually make
sense as a local operator: it is not gauge invariant under small gauge
transformations $\delta A=d\Lambda$.

However we could try to ignore this, and instead simply integrate the whole
thing over a 3-manifold anyway to try and make a conserved charge defect, i.e.
imagine constructing the following object:

$\hat{U}_{{\alpha}}({{\mathcal{M}}}_{3})=\exp\left(i\frac{{\alpha}}{2}\int_{{{\mathcal{M}}}_{3}}\left(\star
j_{A}-\frac{1}{4\pi^{2}}A\wedge dA\right)\right)$ (V.13)

Because we are considering an integral, this is now gauge-invariant under
small gauge transformations $\delta A=d\Lambda$. It is also invariant under
small deformations of ${{\mathcal{M}}}_{3}$ and naively it looks like it forms
a topological charge defect; the first term rotates an operator with integer
axial charge $q$ by ${{\mathcal{O}}}\to
e^{\frac{iq{\alpha}}{2}}{{\mathcal{O}}}$.

The issue however is invariance under large gauge transformations. Recall from
Section IV.6 that this quantizes the coefficients of Chern-Simons terms in the
action; if we want consistency of this object in a situation where
${{\mathcal{M}}}_{3}$ has a non-trivial $1$-cycle, then we require that

${\alpha}\in 2\pi\mathbb{Z}$ (V.14)

This means that we can no longer perform a continuous rotation, i.e. it looks
like there is trivial action on all operators 181818There is a possible
$\mathbb{Z}_{2}$ action depending on whether $q$ is even or odd; this
$\mathbb{Z}_{2}$ is essentially $\psi\to-\psi$, which is part of the $U(1)$
gauge symmetry and is not anomalous..

Thus we cannot form a topological operator on general 3-manifolds
${{\mathcal{M}}}_{3}$, and it really does look like we have lost the axial
symmetry. This may feel somewhat peculiar. In particular, note that the left
hand side has a universal description in terms of the current for the
(preserved) 1-form symmetry $U(1)^{(1)}_{m}$, and one might hope that this
could be given a precise characterization in terms of some sort of
intertwining of these symmetries.

Note also that if we were willing to restrict our attention to topologically
trivial 3-manifolds such as $\mathbb{R}^{3}$ then technically speaking one
could get away with ignoring the issue of large gauge transformations;
nevertheless we might worry that we are missing some key physics.

###  V.2 Charge defect operators for $U(1)_{A}$

The situation changed when the works Choi:2022jqy ; Cordova:2022ieu  explained
how to construct topological charge defects for the ABJ anomaly. Let us begin
by considering only rotation angles that are simple fractions of the form

${\alpha}=\frac{2\pi}{N}$ (V.15)

where $N\in\mathbb{Z}$, which is an arbitrary integer. For the reasons above
$\hat{U}_{{\alpha}}$ does not exist. However let us now consider a different
operator:

$D_{\frac{1}{N}}({{\mathcal{M}}}_{3})\equiv\int[dB]\exp\left[i\int_{{{\mathcal{M}}}_{3}}\left(\frac{2\pi}{2N}\star
j_{A}+\frac{N}{4\pi}B\wedge dB+\frac{1}{2\pi}B\wedge dA\right)\right]$ (V.16)

Here we have done something radical – we have introduced a new dynamical
compact gauge field $B$ and given it a topological Chern-Simons action with
coefficient $N$, and then we have coupled it to the dynamical photon $A$. We
have then done a path integral over that field.

Let me now state some properties of this object. From the first term, it
generates a phase rotation on local operators with axial charge as
${{\mathcal{O}}}\to e^{\frac{2\pi iq}{2N}}{{\mathcal{O}}}$. It is perfectly
well-defined and gauge-invariant since $N$ is in the numerator everywhere.
However if we now try to impose the equations of motion of $B$ we find the
following curious equation:

$B=-\frac{A}{N}$ (V.17)

If you eliminate $B$ using this relation and plug it into the action (V.16)
then you recover – from a well-defined starting point – the previously ill-
defined object (V.13) with a fractional rotation angle
$\hat{U}_{\frac{2\pi}{N}}$!

Thus we have succeeded in obtaining a well-defined topological charge defect
that generates a non-trivial rotation, at the cost of introducing a new field
$a$ on the defect (or equivalently, coupling a TQFT to the bulk degrees of
freedom).

Note that (V.17) does not quite make sense – if you integrate $d$ of it over a
non-trivial cycle you find improperly quantized fluxes. Thus there is a more
rigorous way then what we have just done to show that the defect is
topological, involving gauging a $\mathbb{Z}_{N}$ subgroup of $U(1)_{m}^{(1)}$
on half of the space, but the intuitive argument above gives the correct
answer.

Let us now explore the consequences of this new degree of freedom. Let’s
consider fusing two of these objects with opposite rotation angles:

$D_{\frac{1}{N}}({{\mathcal{M}}}_{3})D_{-\frac{1}{N}}({{\mathcal{M}}}_{3})=\int[dB_{1}][dB_{2}]\exp\left[i\int_{{{\mathcal{M}}}_{3}}\left(\frac{N}{4\pi}B_{1}\wedge
dB_{1}-\frac{N}{4\pi}B_{2}\wedge dB_{2}+\frac{1}{2\pi}(B_{1}-B_{2})\wedge
dA\right)\right]$ (V.18)

This object is not the identity operator, as it would have been for a normal
symmetry – note that the difference gauge field $B_{\Delta}\equiv B_{1}-B_{2}$
has a nontrivial coupling to the photon magnetic field. (In fact the right
hand side is a kind of topological defect called a condensation defect.
Roumpedakis:2022aik ; Choi:2022zal ).

Thus we cannot invert this symmetry any more, and we see that the true effect
of the ABJ anomaly is to turn the ordinary axial symmetry into a non-
invertible symmetry; there is a topological defect associated with it which
counts the charge, but the action of performing a charge rotation can no
longer be undone.

How do we use this object? Its action on normal local operators is
conventional and invertible, and as such it can be used to derive selection
rules on scattering amplitudes and constrain the terms that can appear in RG.
This works the normal way on $\mathbb{R}^{4}$ but can result in interesting
subtleties for other manifolds; see Cherman:2022eml  for a recent discussion.

However its action on non-local objects such as ’t Hooft lines is different.
In general if one drags the ’t Hooft line through one of these objects it
acquires a tail, i.e. it becomes the boundary of a two-dimensional topological
surface which is $\exp\left(\frac{i}{N}\int_{S}F\right)$. This can be
understood as a manifestation of the Witten effect Witten:1979ey . As a
reminder, the Witten effect tells us that if one introduces a topological
coupling ${\theta}$ as ${\theta}\int_{{{\mathcal{M}}}_{4}}F\wedge F$ – which
is essentially what we are doing when we perform the axial rotation above –
then this binds an fractional electric charge charge proportional to
${\theta}$ to a magnetic monopole. A fractional electric Wilson line is not a
well-defined object; the closest one can get to it is to integrate $F$ over a
two-surface $S$ whose boundary can now be imagined to be the Wilson line. This
explanation has been very telegraphic, see Choi:2022jqy  for more details.

Finally, you might feel worried that we have only constructed a rotation angle
by $\frac{2\pi}{2N}$, for any $N$; however clearly by composing this $p$ times
we can obtain any rational fraction $\frac{p}{N}$ of $2\pi$. (In fact there is
a simpler route which uses a different and slightly more involved TQFT from
the simple Chern-Simons theory to arise at the same rational covering of the
unit circle).

The existence of rational numbers may seem peculiar. There is an alternative
route to constructing the charge defects GarciaEtxebarria:2022jky ;
Karasik:2022kkq ; rather than introducing a Chern-Simons field on the defect,
you introduce a compact phase field ${\theta}(x)$ and consider the following
gauge invariant object, parametrized by a continuous $U(1)$ element ${\alpha}$
(and not a rational number).

$U_{{\alpha}}({{\mathcal{M}}}_{3})=\int[d{\theta}]\exp\left(i\frac{{\alpha}}{2}\int_{{{\mathcal{M}}}_{3}}\left(\star
j_{A}-\frac{1}{4\pi^{2}}(A-d\theta)\wedge dA\right)\right)$ (V.19)

Here under a $U(1)$ gauge transformation the scalar $\theta$ transforms as:

$A\to A+d\lambda\qquad\theta\to\theta+\lambda$ (V.20)

This object has the advantage191919It also has a disadvantage: ideally one
would like to be able to orient these charge defects with one dimension in
time and then quantize the theory on an equal-time slice that intersects the
defect. In that case one usually obtains a finite Hilbert space on the defect
interacting with the regular quantum field theory degrees of freedom. The
object (V.19) seems to have an infinite-dimensional Hilbert space living on it
and is thus somewhat ill-defined in comparison to the mathematically precise
degrees of freedom that constitute (V.16). There is something to be resolved
further here; I am grateful to S-H. Shao for discussions on this point. that
it allows one to construct a local density on the defect

$\star\tilde{j}_{A}=\star j_{A}-\frac{1}{4\pi^{2}}(A-d\theta)\wedge dA$ (V.21)

As this object requires the phase field $\theta$ it cannot be defined off the
defect. The existence of this density permits one to prove an analogue of
Goldstone’s theorem for this non-invertible symmetry, which can be viewed as a
perhaps unnecessarily fancy way to understand why $U(1)$ axions are massless
(or, in real life, why the $\pi_{0}$ is so light).

##  VI More applied applications

Here we turn to other applications of higher-form symmetries. Our
considerations up till now have been somewhat formal: though we have studied
some theories which exist in real life, we have not studied them under very
realistic situations. As we will see below, it is in fact quite interesting to
consider the realization of these higher-form symmetries at finite temperature
$T$, providing us with very interesting connections to magnetohydrodynamics
and astrophysics. In this section we assume the reader is familiar with the
approach to finite-temperature quantum field theory in the formalism of the
Euclidean path integral, as described for example in the standard textbooks
kapusta2007finite ; le2000thermal .

###  VI.1 Finite temperature: static phenomena

A very physically relevant example is to study hot QED coupled to massive
electrically charged matter (as in Section IV.3), i.e.

$S[\phi,A]=\int
d^{4}x\left(\frac{1}{4e^{2}}F^{2}+(D\phi)^{\dagger}D\phi+m^{2}\phi^{\dagger}\phi+\cdots\right)$
(VI.1)

If we are interested in static phenomena, we can understand the physics at
finite $T$ by placing the system on $S^{1}\times\mathbb{R}^{3}$, where the
$S^{1}$ has length $\beta=\frac{1}{T}$. Note that the covariant derivative
acts as $D_{\mu}\phi=\partial_{\mu}\phi-iqA_{\mu}\phi$.

Recall that there is generically only a single magnetic 1-form symmetry with
current $J^{\mu\nu}={1\over 2}{\epsilon}^{\mu\nu\rho\sigma}F_{\rho\sigma}$.
Upon dimensional reduction, from the 3d point of view this will break into a
$0$-form symmetry $U(1)^{(0)}_{m}$ with current $j^{i}=J^{i\tau}$ and a
$1$-form symmetry $U(1)^{(1)}_{m}$ with current $J^{ij}$. We should now ask –
as usual – how these symmetries are realized on the $\mathbb{R}^{3}$, i.e. the
non-compact directions.

To understand this, let us try and compute an effective action. Let’s
decompose the vector potential into $A_{\mu}=(A_{\tau},A_{i})$ and understand
what each of the components feels, by integrating out the bosons at finite
temperature. Let us attempt to understand the expected result from elementary
physics, which is that of a gas of charged bosons at finite temperature,
interacting with electric fields. In quantum statistical mechanics, we often
consider bosons at finite chemical potential $\mu$, which can be understood as
the background value of the real-time vector potential $A_{t}=\mu$. Various
quantities typically depend on $\mu$ though a Bose-Einstein distribution of
the form

$n(E)=\frac{1}{e^{\beta(E\pm q\mu)}-1}$ (VI.2)

where the sign depends on whether we are considering the particle or the
antiparticle, which have opposite charge. Now when we go to Euclidean time we
have $A_{\tau}=iA_{t}=i\mu$; let’s also consider a limit where the mass of the
particle is very high, so that $E\approx m$ (i.e. we are ignoring the
contribution from the motion of the particles), and also that $\beta\gg
E^{-1},\mu$ (so that the mass and $\mu$ are high compared to the temperature).

In that case this distribution takes the form

$n(E)\sim e^{-\beta m\pm iq\beta A_{\tau}}$ (VI.3)

Thus if we now imagine constructing an effective action for $A_{\tau}$, the
ingredients take the form shown above, and thus in the limit shown we expect
an answer of the form:

$S[A_{\tau}]=\beta\int
d^{3}x\left(\frac{1}{2e^{2}}(\partial_{i}A_{\tau})^{2}+c_{1}e^{-\beta
m}\cos(q\beta A_{\tau})+\cdots\right)$ (VI.4)

where the first term comes from the bare kinetic term, we have added together
the particle and anti-particle contributions and $c_{1}$ is a non-universal
constant. Note that the answer is periodic in $q\beta A_{\tau}$, as it must be
since a background of the form $q\beta A_{\tau}=2\pi\mathbb{Z}$ around a non-
contractible $S^{1}$ is gauge-equivalent to no background at all.

The reader who is happy with this intuitive argument is free to move on;
however the reader who wants a more concrete derivation may want to derive
this result more formally, by explicitly computing the functional determinant
of the $\phi$ field in the presence of a background constant $A_{\tau}$. To do
this, we denote the eigenvalues of the relevant differential operator by
${\lambda}_{n,\vec{p}}$. They satisfy the following equation:

$(-D^{\mu}D_{\mu}+m^{2})\phi={\lambda}_{n,\vec{p}}\phi$ (VI.5)

and are labeled by an integer $n\in\mathbb{Z}$ describing the dependence on
Euclidean time and a continuous 3-vector momentum $\vec{p}$. Explicitly, they
are:

${\lambda}_{n}=\left(\frac{2\pi
n}{\beta}-qA_{\tau}\right)^{2}+\omega_{\vec{p}}^{2}\qquad\omega_{\vec{p}}^{2}=\vec{p}^{2}+m^{2}$
(VI.6)

and thus we would like to compute the following determinant:

${{\mathcal{L}}}_{\mathrm{eff}}[A_{\tau}]=\log{\rm
det}\left[(-D^{\mu}D_{\mu}+m^{2})\right]=T\sum_{n\in\mathbb{Z}}\int\frac{d^{3}p}{(2\pi)^{3}}\log\left(\left(\frac{2\pi
n}{\beta}-qA_{\tau}\right)^{2}+\omega_{\vec{p}}^{2}\right)$ (VI.7)

The sum over $n$ is divergent, but this UV divergence is unrelated to the
thermal physics we are studying. We can isolate this divergence by taking a
derivative with respect to $\omega_{\vec{p}}$ to find

$\frac{1}{2\omega_{\vec{p}}}\frac{d}{d\omega_{\vec{p}}}\sum_{n}\log\left(\left(\frac{2\pi
n}{\beta}-qA_{\tau}\right)^{2}+\omega_{\vec{p}}^{2}\right)=\sum_{n}\left[\left(\frac{2\pi
n}{\beta}-qA_{\tau}\right)^{2}+\omega_{\vec{p}}^{2}\right]^{-1}$ (VI.8)

Performing the sum over $n$ and integrating over $\omega_{\vec{p}}$ we find
that up to an overall (divergent) $A_{\tau}$-independent additive constant we
have

$\log{\rm
det}\left[(-D^{\mu}D_{\mu}+m^{2})\right]=T\int\frac{d^{3}p}{(2\pi)^{3}}\log\left(\sinh\left({1\over
2}(\beta\omega_{\vec{p}}+iqA_{\tau}\beta)\right)\sinh\left({1\over
2}(\beta\omega_{\vec{p}}-iqA_{\tau}\beta)\right)\right)$ (VI.9)

Let us now work in the limit where the particles are very massive – i.e. where
most of the contribution to the energy comes from the rest mass of the
particle $\omega_{\vec{p}}\approx m$ – and also where this mass is high
compared to the temperature – i.e. $\beta m\to\infty$. Then we can expand the
action in powers of $e^{\frac{-\beta m}{2}}$ and find the leading $A_{\tau}$
dependence to be:

${{\mathcal{L}}}_{\mathrm{eff}}[A_{\tau}]\approx
T\int\frac{d^{3}p}{(2\pi)^{3}}\left(e^{-\beta m}\cos(q\beta
A_{\tau})+\cdots\right)$ (VI.10)

Here we have taken $A_{\tau}$ to be a constant; thus this result is actually
the leading term in a derivative expansion in powers of
$\partial_{i}A_{\tau}$, and the functional form as a function of $A_{\tau}$ is
precisely that shown in (VI.4).

Finally, we can understand this from an even slicker “first-quantized”
language, involving particle worldlines rather than quantum fields. Recall
that the gauge-invariant degree of freedom involving $A_{\tau}$ is the
holonomy of the gauge field around the thermal circle,

$\exp\left(iq\oint_{S^{1}}A_{\tau}\right)\ .$ (VI.11)

In the limit where $A_{\tau}$ is constant in $\tau$ the holonomy becomes
$e^{iq\beta A_{\tau}}$. Particle worldlines – i.e. the quanta of the $\phi$
field – that wrap the circle will feel this holonomy (with the particle
feeling $e^{i\beta qA_{\tau}}$ and the anti-particle feling $e^{-i\beta
qA_{\tau}}$), and each contribution of this worldline will be weighted by
$e^{-\beta m}$ (i.e. the exponential of the particle action, which is just the
length of its trajectory in Euclidean time). Assembling the pieces – and
performing a dilute gas sum over many disconnected particle worldlines that
are each localized in $\mathbb{R}^{3}$, but wrap the $S^{1}$ cycle – we again
find the answer above. Note that this is essentially equivalent to the
mechanism of Polyakov confinement in 3d Polyakov:1976fu ; Polyakov:1975rs ,
with the role of the monopole instanton now being played by the wrapped
electric worldline.

Let us now understand the spatial components $A_{i}$; as it turns out, in the
usual thermal equilibrium these components are essentially unaffected and they
just inherit their bare kinetic term. (In the plasma literature, this is
usually stated as the fact that the “magnetic field is unscreened”).

Assembling the pieces we find the following answer for the full effective
action in 3d:

$S[A_{\tau},A_{i}]=\beta\int
d^{3}x\left(\frac{1}{2e^{2}}(\partial_{i}A_{\tau})^{2}+\frac{1}{4e^{2}}(\partial_{i}A_{j})(\partial^{i}A^{j})+c_{1}e^{-\beta