8. Quantum Field Theory on the Plane
In this section, we step up a dimension. We will discuss quantum field theories in
d = 2 + 1 dimensions. Like their d = 1 + 1 dimensional counterparts, these theories
have application in various condensed matter systems. However, they also give us
further insight into the kinds of phases that can arise in quantum field theory.
8.1 Electromagnetism in Three Dimensions
We start with Maxwell theory in d = 2 = 1. The gauge field is Aµ , with µ = 0, 1, 2.
The corresponding field strength describes a single magnetic field B = F12 , and twoelectric fields Ei = F0i . We work with the usual action,
Z
1
SMaxwell = d3 x
Fµ⌫ F µ⌫ + Aµ j µ
(8.1)
4e2
The gauge coupling has dimension [e2 ] = 1. This is important. It means that U (1)gauge theories in d = 2 + 1 dimensions coupled to matter are strongly coupled in theinfra-red. In this regard, these theories di↵er from electromagnetism in d = 3 + 1.
We can start by thinking classically. The Maxwell equations are1@µ F µ⌫ = j ⌫e2Suppose that we put a test charge Q at the origin. The Maxwell equations reduce tor2 A0 = Q 2 (x)which has the solution
Q
A0 =
log
2⇡

✓

r
r0

◆

+ constantfor some arbitrary r0 . We learn that the potential energy V (r) between two charges,
Q and Q, separated by a distance r, increases logarithmically✓ ◆Q2rV (r) =
log
+ constant
(8.2)
2⇡
r0
This is a form of confinement, but it’s an extremely mild form of confinement as thelog function grows very slowly. For obvious reasons, it’s usually referred to as log
confinement.

– 387 –

In the absence of matter, we can look for propagating degrees of freedom of the gauge
field itself. As explained in the previous section, we expect the gauge field to have asingle, propagating polarisation state in d = 2 + 1 dimensions.
8.1.1 Monopole Operators
Something special happens for U (1) gauge theories in d = 2 + 1 dimensions: they
automatically come an associated global U (1) symmetry that we will call U (1)top , the“top” for “topological”. The associated current is
µ
Jtop
=

1 µ⌫⇢
✏ F⌫⇢
4⇡

(8.3)

µ
which obeys the conservation condition @µ Jtop
= 0 by the Bianchi identity on Fµ⌫ . The
associated conserved quantity is simply the magnetic flux
Z
Z
1
2
0
Qtop = d x Jtop =d2 x B
2⇡

In quantum field theory, symmetries act on local operators. The operators that transform under U (1)top are not the usual fields of the theory. Rather, they are disorder
operators, entirely analogous to the ’t Hooft lines that we introduced in Section 2.6. In
the present context, they are referred to as monopole operators.
We work in Euclidean space. A monopole operator M(x) inserted at a point x 2 R3
is defined in the path integral by requiring that we integrate over field configurations
in which there is a Dirac monopole inserted inserted at the point x. This means that,for an S2 surrounding x, we haveZ
1
d2 Sµ ✏µ⌫⇢ F⌫⇢ = 1
(8.4)
4⇡ S2
This operator creates a single unit of magnetic flux so that, in the presence of M(x),the topological current is no longer conserved; instead it has a sourceµ@µ Jtop
=

3

(x)

(8.5)

Equivalently, the monopole operator is charged under U (1)top so that
U (1)top : M(x) 7! ei↵ M(x)(8.6)

The definition of monopole operators given above is somewhat abstract. As we will
now see, in certain phases of the theory it is possible to give a more concrete definition.

– 388 –

Consider free Maxwell theory. Alternatively, consider U (1) gauge theory coupled tocharged fields with masses m
e2 . In both cases, the theory lies in the Coulomb phase,
meaning that low energy spectrum contains just a single, free massless photon. The
partition function is particularly straightforward; ignoring gauge fixing terms, we have
✓ Z
◆
Z
1
3
µ⌫
Z = DAµ exp
dx
Fµ⌫ F
4e2
Because the action depends only on Fµ⌫ , and not explicitly on Aµ , we can choose
instead to integrate over the field strength. However, we shouldn’t integrate over all
field strengths; in the absence of monopole operators, we should integrate only overthose that satisfy the Bianchi identity ✏µ⌫⇢ @µ F⌫⇢ = 0. We can do this by introducing a
Lagrange multiplier field (x),
✓ Z
◆
Z
1
i µ⌫⇢
3
µ⌫
Z = DFµ⌫ D exp
dx
Fµ⌫ F +
✏ @µ F⌫⇢
(8.7)
4e2
4⇡
If the field strength obeys the Dirac quantisation condition, then has periodicity 2⇡.
But in this formulation, it is particularly straightforward to implement a monopole
operator. We simply add to the path integral
M(x) ⇠ ei (x)(8.8)

This ensures that the topological current has a source (8.5) or, equivalently, inserts amonopole at x.
We can now go one step further, and integrate out the field strength Fµ⌫ . We’re left
with an e↵ective action for the Lagrange multiplier field (x) which, in this context, isusually referred to as the dual photon. We’re left with the e↵ective action,
✓ Z
◆
e2
3
µ
Z = exp
dx
@µ @
(8.9)
8⇡ 2
Clearly this describes a single, propagating degree of freedom. But this is what we
expect for a photon in d = 2 + 1 which has just a single polarisation state.
In this formulation, the global symmetry U (1)top is manifest, and is given by
U (1)top :7!

+↵(8.10)

This agrees with our expected symmetry transformation (8.6) given the identification(8.8). The associated current can be read o↵ from (8.9); it is
µ
Jtop
=

e2
@µ
2
(2⇡)

– 389 –

(8.11)

There’s one, nice twist to this story. The theory (8.9) has a degeneracy of ground states,
given by constant 2 [0, 2⇡). These degenerate ground states reflects the fact that if
we place some magnetic flux in the Coulomb phase then it spreads out. In any of theseground states, the global symmetry U (1)top acts like (8.10) and so is spontaneously
broken. The associated Goldstone boson is simply itself. But this is equivalent to
the original photon. We have the chain of ideas
Coulomb Phase : Unbroken U (1)gauge,,

Spontaneously Broken U (1)top
Goldstone Mode = Photon

A related set of ideas also holds in higher dimensions, but now with the U (1)top a
generalised symmetry, which acts on higher dimensional objects, as we discussed in
Section 3.6.2. d = 2 + 1 dimensions is special because the disorder operator M(x) isa local operator, ensuring that U (1)top is a standard global symmetry, rather than theless familiar generalised symmetry.
8.2 The Abelian-Higgs ModelWe can get some more intuition for the role of monopole operators, and 3d gauge
theories in general, by looking at the Abelian-Higgs model. This is a U (1) gauge
theory coupled to a scalar field which we take to have charge 1. The action isZ
1 2
SAH = d3 x
F + |Dµ |2 m2 | |2
| |4
(8.12)
4e2 µ⌫
2

We will look at what happens to this theory as we vary the mass m2 from positive
to negative. This is a game that we’ve already played in both d = 3 + 1 dimensions(in Section 2.5.2) and in d = 1 + 1 dimensions (in Section 7.2). In both cases, theinteresting physics came from vortices in the m2 < 0 phase, and the same will be true
here.
When the mass is small, |m| ⌧ e2 , the theory is strongly coupled in the infra-red.
It is difficult to get a handle on the physics here, although we will ultimately be able
understand what happens. In contrast, when |m|e2 , we can first understand the
dynamics of the scalar in a regime where the gauge field is weakly coupled, and thenfigure out what’s left. We first look at these two phases.
m2
e4 : When m2 > 0 we can simply integrate out the scalar, to leave ourselveswith free Maxwell theory below the scale of m2 . This is the gapless Coulomb phase, inwhich we have an unbroken U (1) gauge symmetry. As we explained above, this meansthat the global symmetry U (1)top is spontaneously broken. The Goldstone mode is thephoton.

– 390 –

There are also massive, charged excitations in this phase that come from the field.
They interact through the Coulomb force which means that charges of opposite signexperience a logarithmically confining potential (8.2).
m2 ⌧e4 : This is the Higgs phase. The scalar condenses,| |2 =

m2

giving the photon a mass. This phase is gapped. The U (1)gauge symmetry is spontaneously broken. But now the global topological symmetry U (1)top is unbroken.
The finite energy states of the theory which carry non-vanishing Qtop charge are
the vortices. We discussed these in detail in both d = 3 + 1 dimensions where thevortices are strings (see section 2.5.2) and in d = 1 + 1 dimensions where the vortices
are instantons (see Section 7.2). In d = 2 + 1, vortices are particle-like excitations.
They are classical configurations in which the phase of winds asymptotically in thespatial plane R2 . They have finite energy, and finite quantised magnetic flux
I

i

dx @i

1
=
2⇡

Z

d2 x B = Qtop 2 Z

This is what monopole operators do in the Higgs phase: they create vortices. The
upshot is that we can characterise the Higgs phase of the theory as
Higgs Phase : Spontaneously broken U (1)gauge,,

Unbroken U (1)top
Charged Excitation = Vortex

m2 = 0: In d = 2 + 1, the two phases at m2 > 0 and m2 < 0 are clearly di↵erent
since they have a di↵erent global symmetry U (1)top . (This is in contrast to the story
in d = 1 + 1 where vortices are instantons and blur the distinction between the twophases.)We can ask: what happens as we dial m2 from positive to negative. We expect aphase transition to occur at some point, we which we heuristically refer to as m2 = 0.
(In practice, this point can be shifted away from zero). Is this a first order phasetransition, or second order? If second order, what universality class does the theory
lie in? Because the theory is strongly coupled in the regime |m| . e2 it is difficult toperform any quantitative calculations to answer this question. Instead, we will guess.

– 391 –

To guide our guess, we use the symmetries of the problem. Since we have identified
the global U (1)top symmetry as distinguishing phases, it seems reasonable to postulatethat the phase transition lies in the same universality class as other theories governedby a U (1) global symmetry. This turns out to be true, and underlies a rather beautiful
feature of 3d gauge theories known as particle-vortex duality.
8.2.1 Particle-Vortex Duality
In quantum field theories, there are very often two kinds of particle excitations that canappear. The first kind is the familiar excitation that we get when we quantise a local
field. This is that kind that we learned about in our Quantum Field Theory course.
The second kind we’ve seen a number of times in these lectures: they are solitons.
Despite the fact that these two kinds of particles arise in di↵erent ways, there isreally little di↵erence between them in the quantum theory. In particular, both aredescribed as states in the Fock space. Typically at weak coupling, the solitons aremuch heavier than the “elementary particles”, but that’s more a limitation of our need
to work at weak coupling. It may be – and often is – that as we move into strongly
coupled regimes, the solitons become light.
This opens up an intriguing possibility. Is it possible to write down a di↵erent
quantum field theory in which the roles of solitons and elementary particles are reversed.
These two quantum field theories would describe the same physics, but what appears
as a soliton in one would appear as an elementary particle in the other, and vice versa.
This is referred to a duality.
In fact, we’ve already met a simple example of a duality in these lectures. In Section
7.5, we used bosonization to demonstrate the equivalence between a massive fermion
and the Sine-Gordon model. The elementary fermion arises as a kink in the Sine-Gordon
model.
Typically, dualities get harder to construct with any conviction as the number ofdimensions increases. There wonderful examples of dualities in d = 3 + 1, which exchange electric and magnetic excitations, but they need supersymmetry to keep control
over the dynamics and so are beyond the scope of these lectures. However, things are
somewhat easier in d = 2 + 1. Here we do have examples of dualities. In contrast to
the bosonization story of Section 7.5, we are unable to prove the d = 2 + 1 dualities
from first principles, but nonetheless have convincing evidence that they are true. We
will see a number of these dualities as we proceed.

– 392 –

As we’ve seen above, in d = 2 + 1 dimensions the appropriate solitons are vortices.
We will now propose a second theory, whose classical dynamics is di↵erent from theAbelian-Higgs model (8.12), but whose quantum dynamics is argued to be identical.
The vortices in one theory are identified with the elementary particles of the other. For
this reason, the claimed equivalence of the two theories is referred to as particle-vortex
duality.
The XY-Model
The theory which is claimed to be dual to the 3d Abelian-Higgs model is simply atheory of a complex scalar field ˜, without any gauge field,
SXY =

Z

d3 x |@µ ˜|2m̃2 | ˜|2

˜
2

| ˜|4

(8.13)

This is known as the XY-Model. At first glance, the physics of this model is ratherdi↵erent from the XY-model. Indeed, at first glance it appears to have fewer degrees
of freedom because it is missing the gauge field. Nonetheless, as we now explain, theydescribe the same physics, albeit in a non-obvious and interesting way.
Let’s first address the issue of degrees of freedom. The XY-model clearly has twodegrees of freedom in the UV where it is weakly coupled. But the Abelian-Higgs modelhas the same number: the gauge redundancy removes one degree of freedom from ,but this is replenished by the single polarization state of the photon. We learn aninteresting lesson: gauging a U (1) symmetry in d = 2 + 1 changes the dynamics, butdoes not change the overall number of degrees of freedom. This will be important inlater developments.
We can also match the symmetries between the XY-model and the Abelian-Higgs
model. The XY-model clearly has a U (1) global symmetry which rotates the phase of
˜. The associated current is⇣⌘
J µ = i ˜† @ µ ˜ (@ µ ˜† ) ˜
XY

The Abelian-Higgs model also has a single global symmetry that we called U (1)top . Youmight worry that the Abelian-Higgs model also has a gauge symmetry, which is clearly
not shared by the XY-model. But, as we have stressed many times, gauge symmetries
are not symmetries at all, but redundancies. This gives another important lesson: there
is no need for gauge symmetries to match on both sides of a duality.
We can now look at how the physics of the XY-model changes as we vary the mass:– 393 –

m̃2 > 0: This is a gapped phase. The ˜ excitations are massive and carry charge
under the unbroken U (1) global symmetry. We see that, at least with broad brush, thislooks similar to the the Higgs phase of the Abelian-Higgs model, in which the U (1)top
symmetry was unbroken. In that case, the vortices carried charge under U (1)top .
m̃2 < 0: In this phase, ˜ gets a vacuum expectation value and the U (1) global
symmetry is broken. We can write ˜ = ⇢ei . The fluctuations of ⇢ are massive, while
the field is massless: it is the Goldstone mode for the broken U (1). Notice that we’ve
given this field the same name as the dual photon in the Abelian-Higgs model. This is
not a coincidence.
Again, with broad brush this looks similar to the gapless Coulomb phase of the
Abelian-Higgs model. However, the Coulomb phase was also characterised by theexistence of massive, charged excitations that were logarithmically confined. Can wesee similar excitations in the XY-model? The answer is yes.
The ordered phase of the XY-model also has vortices. As before, these arise fromthe phase of ˜ winding asymptotically, but now there is no gauge field to cancel thelog divergence in their energy,
Z
Z
Z 1
1
n2 ˜ 222
2
˜˜d x |@i | = d✓dr r 2 |@✓ | + . . . = 2⇡
dr
| | + ...
rr
0
The energy of a single vortex is logarithmically divergent. But this divergence canbe cancelled by placing an anti-vortex at some distance r. It’s not hard to convince
yourself that the logarithm reappears in the potential energy between the vortex andanti-vortex, which scales as
✓ ◆
1
r
V =
log
2⇡
r0
for some cut-o↵ r0 . In other words, the vortices are logarithmically confined. This, ofcourse, is the same behaviour exhibited by charged particles in 3d electromagnetism.
m̃2 = 0: Lying between the two phases above is a critical point. Once again, we arebeing a little careless in describing this as sitting at m̃ = 0; strictly, you should tune
both m̃ and the other parameters to hit the critical point.
This time, the physics of the critical point is well understood: this is the XY WilsonFischer fixed point. We studied this in some detail in the lectures on Statistical Field
Theory using the epsilon expansion.

– 394 –

The essence of particle-vortex duality is the claim that the Abelian-Higgs model also
flows to the XY Wilson-Fisher fixed point at m = 0. This claim can be traced back towork of Peskin in the 1970s, but was brought to prominence by Dasgupta and Halperinin the early 1980s. Given the similarity in their phase structure, this would seem tobe a reasonable claim. There is currently no proof of the duality, but there is now
convincing numerical evidence that it is true.
The Duality Dictionary
The key to particle-vortex duality is really the idea of universality: the two theories(8.12) and (8.13) share the same critical point. We can then attempt to map the operators of the two theories at the critical point. We have only an incomplete dictionary
at the moment, but our discussion above allows us to start to fill in some entries. For
example, we have seen how the currents match on both sides
µ
Jtop

µ
JXY

!

With two theories flowing to the same critical point, we can now turn on relevant
operators in each. As long as we turn on the same relevant operator, we are guaranteedthat the theories coincide in the neighbourhood of the fixed points. We have seen above
how this plays out: when the scalar condenses in one theory, it matches the phase inwhich the scalar is not condensed in the other. Roughly speaking, we have
m2 ⇡

m̃2

Alternatively, we can write this in terms of the relevant operators at the critical point
as
| |2

!

| ˜|2

(8.14)

although since the critical points are strongly coupled, this relation is likely to havecorrections, with operators on both sides mixing with others.
Far from the critical point, we have seen that the theories have the same qualitative
features. In particular, the duality inherits its name from the map between massive
excitations,gauge vortex
excitation!
!

˜ excitation
global vortex

Only the first of these describes a map between finite energy excitations. In this case, it
is better to phrase the map in terms of local operators, rather than solitons: the essence– 395 –

of particle-vortex duality is that the monopole operator on one side is a traditional fieldin the Lagrangian on the other,
M(x)

!

˜(x)

(8.15)

We could ask: do the interactions between these massive excitations agree in detail?
The answer is most likely no. One could add irrelevant operators to both the AbelianHiggs model and the XY model which will a↵ect the interactions between these massive
particles. We would have to work much harder to get quantitative agreement awayfrom the critical point. For what it’s worth, it is possible to do this matching in certain
supersymmetric versions of the duality. Here, particle-vortex duality is referred to as
3d mirror symmetry.
The View from Statistical Physics
The claim of particle-vortex duality o↵ers a very clear experimental prediction. Although we have phrased our discussion in the context of physics in d = 2+1 dimensions,
everything goes through in the the Euclidean d = 3 + 0 world. Here, the theories (8.12)
and (8.13) can be viewed as statistical field theories, with the path integral describing
thermal rather than quantum fluctuations. More details can be found in the lecturenotes on Statistical Field Theory.
In this context, the 3d XY-model (8.13) governs the phase transition of a number ofsystems, including the superfluid transition of liquid helium. Similarly, the 3d AbelianHiggs model (8.12) governs the superconducting phase transition, with the field strength
Fij , i, j = 1, 2, 3 describing the fluctuating magnetic field.
In both cases, the mass2 term determines the deviation from the critical temperature
Tc at which the phase transition occurs. But that makes the map (8.14) between themasses rather surprising. It means that the duality maps the high temperature phaseof the superfluid to the low temperature phase of the superconductor, and vice versa.
The claim that both theories share a critical point then becomes the claim that thetwo phase transitions have the same critical exponents. Experimentally, however, thisclaim is incorrect: the two phase transitions are not the same. While the superfluid
transition exhibits the XY Wilson-Fisher exponents, the superconducting transition
has mean field exponents. It would seem that particle-vortex duality has been ruledout experimentally!

– 396 –

In fact this is too quick. Recall that the XY-model has two critical points. The mean
field critical point is unstable, with | |4 a relevant operator that drives the theory to
the Wilson-Fisher point. The same should be true of the Abelian-Higgs model. It isthought that the mean field exponents seen in the superconducting transition reflect
the fact that the experiments haven’t got close enough to the true critical point, andare instead probing the unstable mean field point. Calculations suggest that one would
start to see Wilson-Fisher critical exponents in the superconducting transition only at
T Tc ⇠ 10 9 K. Such a level of precision is not technologically feasible.
But this brings its own issues. It appears that we have a system in Nature which
is fine-tuned. The natural scale of the superconducting phase transition is Tc ⇠ 10 K
or so. In the experiments, we tune the coefficient of | |2 by hand to hit the criticaltemperature. But why is the coefficient of the | |4 relevant operator so small that it
only shows up when T Tc ⇠ 10 9 K? This is similar to the famous hierarchy problem
in the Standard Model, where again the coefficient of a relevant operator appears to
be fine-tuned.
Particle physicists have sleepless nights over fine tuning, and desperately search foran explanation. In large part, this is because of experience with RG in statisticalphysics, where any fine-tuning seen in Nature must also have an explanation. In thecase of superconductors, the apparent fine tuning is understood: it arises because theunderlying scalar field is not fundamental, but instead comprises of a Cooper pairof electrons. (The analogous possibility for the Higgs fine tuning goes by the nameof technicolour.) A full explanation would take us too far from the purpose of theselectures, but this suffices to ensure that the smallness of the | |4 relevant operator seenin the superconducting transition is technically natural.
8.3 Confinement in d = 2 + 1 Electromagnetism
We’ve seen that classical electromagnetism in d = 2 + 1 dimensions confines particles,
but only weakly with a log potential✓ ◆Q2r
V (r) =
log2⇡
r0
There is, however, an important e↵ect in the quantum theory that turns the logarithmic
confining potential into a more powerful linearly confining potential. This e↵ect, first
discovered by Polyakov, is due to instantons.
We’ve met instantons in d = 3 + 1 Yang-Mills theory in Section 2.3, and again in
the d = 1 + 1 Abelian-Higgs model in Section 7.2. In the latter case, vortices that play

– 397 –

the role of instantons. Now that we are living in d = 2 + 1 dimensions, the instantons
should be objects localised in three Euclidean dimensions. But these are very familiar:
they are magnetic monopoles.
We’ve already introduced the idea of monopole operators in Section 8.1. These canbe thought of Dirac monopoles at a point. They are not quite what we want for thepresent purposes. As a starting point for a semi-classical calculation, we would like themonopoles to be smooth configurations with finite action. But we’ve seen such objects
before: we can use the ’t Hooft Polyakov monopole described in Section 2.8.
Recall that the ’t Hooft Polyakov monopoles arise in an SU (2) gauge theory (or,more generally, any non-Abelian gauge theory) broken down to its Cartan subalgebra.
To achieve this, we couple the SU (2) gauge theory to a real, adjoint scalar and workwith the action
✓
◆2
Z
1
1
v2
3
µ⌫
2
2
S= dx
tr Fµ⌫ F + 2 tr (Dµ )tr(8.16)
2g 2
g
4
2
The ground state of the system has, up to a gauge transformation,
the gauge symmetry= v 3 , and breaks

SU (2) ! U (1)At low energies, the spectrum contains just a single massless photon andp looks likepure electromagnetism. In addition, there is a neutral scalar with mass ⇠gv and acharged W-boson of mass ⇠ v.
In this way, we can view the model as U (1) gauge theory, with a UV cut-o↵ at thescale v. The dimensionless gauge coupling constant is g 2 /v and to trust any semiclassical calculation, we must take g 2 /v ⌧ 1.
8.3.1 Monopoles as Instantons
Our main reason for introducing the action (8.16) is that, in Euclidean spacetime, it
admits smooth monopole solutions. These are the ’t Hooft Polyakov monopoles that
we introduced in Section 2.8, but now localised in Euclidean spacetime meaning thatthey play the role of instantons, rather than particles. Here we recount the basics.
The existence of the monopoles can be traced to topology. Any finite action configuration must obey tr 2 ! v 2 as x ! 1. This defines a sphere S2 in field space, so all
finite action configurations are classified by a winding number ⇧2 (S2 ) = Z, defined asZ1⌫=d2 Si ✏ijk ✏abc a @j b @k c 2 Z(8.17)
8⇡v 3 S21

– 398 –

However, winding comes at a cost. Any purely scalar configuration that winds has
linearly divergent action. This can be compensated by turning on a gauge field andthis, in turn, endows the soliton with magnetic charge in the unbroken U (1) ⇢ SU (2),(2.91),
Z
1
1m=
d2 Si ✏ijk tr (Fjk ) = 4⇡⌫v
2
The solution for a single monopole, with winding ⌫ = 1, has asymptotic formxa
xj
and Aai ! ✏aij 2 as x ! 1rr
The action of this configuration is finite, and given by
8⇡v
Smono = 2 f ( g 2 )ga!v

with f ( g 2 ) a monotonically increasing function. It has the property that f (0) = 1, sothat the action above coincides with that of a BPS monopole (2.93) when = 0.
We’re used to the idea that finite action configurations in Euclidean space tunnel
between di↵erent vacua of the theory. But what vacua does the monopole tunnelR
between? Clearly, it changes the magnetic flux = d2 x B on a spatial slice. If wewere living on a compact space, this would change the energy of a state, which is given
by
✓
◆2
Z
1 2 1
2
E = d x B ⇠ Area
2
2
Area
with “Area” the area of a spatial slice. However, as the area tends to infinity, the flux
is suitably diluted and the cost in energy is vanishingly small. These are the di↵erent
vacua that the monopoles tunnel between.
A Dilute Gas of Monopoles and Anti-Monopoles
With our monopole solution in hand, we can use it as the starting point for a semiclassical evaluation of the path integral. We should be getting used to this by now, andwe follow the structure of the calculation laid out in Section 2.3, and again in Section
7.2.
One key step in the calculation is to invoke the use of a dilute gas of instantons.
In the present case, this means we treat configurations of widely separated monopoles
and anti-monopoles, with magnetic charges mi = ±4⇡, as saddle points in the path
integral. In the previous situations, we argued that the action of a dilute gas of N
(anti)-instantons was roughly S ⇡ N Sinst , reflecting the fact that these are approximate
solutions when the objects are far separated.

– 399 –

For monopoles, however, we should treat this step more carefully. Viewed as particles
in d = 3 + 1 dimensions, we know that the energy will pick up contributions from thelong range Coulomb forces between the monopoles. This translates into a contribution
to the action in our context. If a monopole of charge mi = ±4⇡ sits at position Xi , thetotal action will be
X ⇣ m i ⌘2
1 X mi mj
S = Smono+4⇡4⇡g 2 i6=j |Xi Xj |iwhere the second term reflects the long range Coulomb interaction.
We evaluate the path integral by summing over these dilute gas configurations, containing N constituents of either type. This results in the expression,
!
Z Y
1
N
X
X 1
X
1
m
m
N
i j
Z=
Ke Smonod3 Xi exp
(8.18)
2
i
j|
N
!
8⇡g
|X
X
i=1
N =0 m =±4⇡i6=jiHere K is the usual contribution from one-loop determinants and Jacobian factors. We
could compute it, but it does not give any qualitatively new insights into the physics
so we will not. The second factor in the expression above is the novelty. When the
instantons are non-interacting, this just gives a power of V N to the path integral,with V the spacetime volume. Now that we have long range interactions between the
instantons, we must work a little harder.
There is a useful way to rewrite the final expression. We use the fact that the 1/r
factor also arises in the Green’s function of the Laplacian in three dimensions. In
general, for a scalar field (x), and any fixed function f (x), we have
✓ Z
◆
✓ Z
◆
Z
1
1
3
2
3
3 f (x)f (y)D exp
d x (@µ ) + f (x) (x) ⇠ exp
d xd y2
8⇡|x y|
Using this, we rewrite the sum over the Coulomb gas in (8.18) as a path integral! Z
Z
1 X mi mj
g2
i X
3
2
exp
=
D
exp
d
x
(@
)
+
mi 3 (xµ2i
j28⇡g i6=j |X
X |
8⇡
2⇡ i

Xi )

(We used a very similar trick in the lectures on Statistical Field Theory when treating
the 2d Coulomb gas in the XY model.)In fact, we’ve met this field (x) before: it is precisely the dual photon that we
introduced in Section 8.1. To see this, note that the coupling to the magnetic charge
above coincides with the coupling in (8.7)– 400 –!

Continuing with our calculation, the partition function becomes
Z =

Z

=

Z

=

Z

D

D
D

exp

✓ Z

exp

✓ Z

exp

✓ Z

g2
dx
(@µ )2
2
8⇡

◆ X
1

g2
dx
(@µ )2
2
8⇡

◆ X
1

✓

Ke

Smono

3

3

g2
dx
(@µ )28⇡ 2
3(Ke Smono )N
N!
N =0
Z Y
N
X
i P
⇥
d 3 Xie 2⇡ i mi (Xi )
1
N!
N =0

mi =±4⇡

i=1

Ke

Smono

cos(2 )◆

Z3d x cos(2 (x))◆N
(8.19)

We can now see the net e↵ect of the instantons: they have generated a potential forthe dual photon . Expanding about the minimum at = 0, we find that the dual
photon has acquired a mass,
m2photon =4⇡ 2 Ke Smonog2

On dimensional grounds, the one-loop determinants and Jacobian factors that we
lumped into the constant K must have dimension [K] = 3. For small , it turns
out to scale as K ⇠ v 7/2 /g. At weak coupling g 2 /v ⌧ 1 and Smono
1, where oursemi-classical analysis is valid, we find that the mass of the dual photon is exponentiallysmaller than all other scales in the game. This means that we can read o↵ the e↵ective
action from (8.19)
Se↵ =

Z

d3 x

g2
(@µ )2 + Ke Smono cos(2 )8⇡ 2(8.20)We recognise this as the Sine-Gordon model that we met in d = 1 + 1 dimensions in
Section 7.5.5. Now it arises as the e↵ective, low-energy description of a gauge theory
in d = 2 + 1 dimensions.
8.3.2 Confinement
What does it mean for the dual photon to get a mass? To answer this, we can see how
the ground state responds to various provocations.
First, let’s try to turn on an electric field in the ground state, say F01 6= 0. To
understand what this means in terms of the dual photon, we need to relate Fµ⌫ with– 401 –

. We can do this by comparing our expressions for the topological current (8.3) and(8.11),
µ
Jtop
=

1 µ⌫⇢
e2 µ
✏ Fµ⌫ =@
4⇡
(2⇡)2

We find that an electric field corresponds to
F01 =

e2
@2
2⇡

However, the configuration @2 = constant does not obey the equations of motion of
our e↵ective action (8.20). This means that the vacuum does not support a constant,background electric field. Instead, solutions to the equations of motion with @2 6= 0
are kinks, or domain walls, in which interpolates from, say, = 0 as x2 ! 1, to= 2⇡ as x2 ! +1. We already met these kinks in Section 7.5.5 when discussing theSine-Gordon model in d = 1 + 1 dimensions. In the present context, the domain walls
are string-like configurations stretched in the x1 direction, with width ⇠ 1/mphoton in
the x2 direction, and tension,=

4p
2Kg 2 e Smono⇡a result which follows from translating our earlier result (7.69). (Up until now, we’ve
always referred to the string tension as . Obviously that’s a bad choice for our current
discussion.)
The domain wall, or string, is a collimated flux tube of electric field F01 6= 0. This is
the expected behaviour of a gauge theory that is linearly confining. In other words, theclassical log potential (8.2) of 3d gauge theories has been replaced with a more severe,
V (r) = r
We could explicitly compute the Wilson loop in this framework and confirm that it
does indeed exhibit an area law.
We have seen that 3d electromagnetism exhibits linear confinement due to instantons
which, in this context, are monopoles. It is crucial that these monopoles have a finite
action, which we achieved by embedding the theory in a non-Abelian gauge group. If
we introduce other UV completions of the theory, with a finite cut-o↵, ⇤U V , these too
will have monopoles, typically with action Smono ⇠ ⇤U V /g 2 . (Lattice gauge theory
provides a good example of this). These too will then exhibit linear confinement.

– 402 –

8.4 Chern-Simons Theory
Gauge theories in d = 2 + 1 dimensions admit a rather special interaction that does
not have a counterpart in even spacetime dimensions. This is the famous ChernSimons interaction. It plays a key role in many areas of theoretical and mathematicalphysics, from the physics of the quantum Hall e↵ect, to the mathematics of the knot
invariants. Many details on the former application can be found in the lecture noteson the Quantum Hall E↵ect.
For U (1) gauge theory, the Chern-Simons term takes the form
Z
k
SCS =
d3 x ✏µ⌫⇢ Aµ @⌫ A⇢
4⇡(8.21)

We could consider this term on its own, or in conjunction with the Maxwell action(8.1). In either case, the dimensionless coefficient k is known as the level. We canwrite down similar terms in any odd spacetime dimension; we briefly met the d = 4 + 1
dimensional version in Section 4.4.1.
Let’s start by studying the symmetries of the Chern-Simons action. It is Lorentz
invariant, courtesy of the ✏µ⌫⇢ invariant tensor. At an operational level, the existenceof this tensor means that the term is exclusive to d = 2 + 1 dimensions. However,this same ✏µ⌫⇢ tensor means that the Chern-Simons interaction breaks both parity andtime-reversal invariance. Here we focus on parity. In even dimensions we can always
take parity to act as x 7! x (see, for example, (1.25)). But, in odd dimensions, thiscoincides with a rotation. We should instead take parity to flip the sign of just a single
spatial coordinate,x0 ! x0

,

x1 !

x1

,

x2 ! x 2(8.22)

and, correspondingly, A0 ! A0 , A1 ! A1 and A2 ! A2 . This means that, as
advertised, the Chern-Simons action is odd under parity.
8.4.1 Quantisation of the Chern-Simons levelAt first glance, it’s not obvious that the Chern-Simons term is gauge invariant since it
depends explicitly on Aµ . However, under a gauge transformation, Aµ ! Aµ + @µ !, wehaveZkSCS ! SCS +d3 x @µ (!✏µ⌫⇢ @⌫ A⇢ )4⇡
The change is a total derivative. In many situations we can simply throw this total
derivative away and the Chern-Simons term is gauge invariant. However, there are– 403 –

some situations where the total derivative does not vanish. As we will now show, inthese cases the Chern-Simons partition function is gauge invariant provided that
k2Z

(8.23)For Abelian Chern-Simons theories, it’s a little subtle to see the requirement (8.23)
since it only shows up in the presence of magnetic flux. (This is to be contrasted withthe situation for non-Abelian Chern-Simons theories described in Section 8.4.3 where
one can see the analogous quantisation condition around the vacuum state.)
Perhaps the simplest way is to consider the theory on Euclidean spacetime S1 ⇥ S2 .
We then add a single unit of magnetic flux through the S2 . As we’ve seen many times
in these lectures, if we take the gauge group to compact U (1), the flux is quantised, inthe minimal unitZ1
F12 = 1
(8.24)
2⇡ S2
We then consider large gauge transformations of this background that wind aroundthe S1 . We denote the radius of this S1 as R, and parameterise it by the coordinatex0 2 [0, 2⇡R). Consider a gauge transformation Aµ ! Aµ + @µ ! which winds aroundthe S1 , with
!=

x0
R

(8.25)

Under such a transformation, any matter field with charge q 2 Z remains single
valued, since ! eiq⌧ /R . Even in the absence of charged matter, the statement that
we’re working with a compact U (1) gauge group, rather than a non-compact R gauge
group, means that the theory admits fluxes (8.24) and gauge transformations (8.25).
Under the gauge transformation (8.25), we haveA0 ! A0 +1
R(8.26)

This means that the zero mode of A0 is a periodic variable, with periodicity 1/R.
(We came to the same conclusion in Section 7.1 where we discussed two dimensional
electromagnetism on a spatial circle.)
We can now see what becomes of our Chern-Simons action under such a gaugetransformation? Evaluated on a configuration with constant A0 , we have
Z
k
SCS =
d3 x A0 F12 + A1 F20 + A2 F01
4⇡– 404 –

Now it’s tempting to throw away the last two terms when evaluating this on our background. But we should be careful as it’s topologically non-trivial configuration. We cansafely set all terms with @0 to zero, but integrating by parts on the spatial derivatives
we get an extra factor of 2,
Z
k
SCS =
d3 x A0 F12(8.27)
2⇡
Evaluated on the flux (8.24), with constant A0 = a, we have
SCS = 2⇡kRa
And under the gauge transformation (8.26), we have
SCS ! SCS + 2⇡k
The Chern-Simons action is not gauge invariant. But all is not lost. The partition
function depends only on eiSCS and this remains gauge invariant provided k 2 Z, which
is our claimed result. This last part of the argument is exactly the same as the one we
met in Section 2.1.3 when we discussed Chern-Simons terms in quantum mechanics,
and in a number of other places when we’ve discussed WZW terms.
Chern-Simons Theories and Spinors
There are further subtleties associated to the factor of 2 above, which we flag up here.
A better way to think about the Chern-Simons theory on a 3-dimensional manifold M ,is by viewing this as the boundary of 4-dimensional manifold X. The story is simplest
in the language of forms, where we haveZZkk
SCS [A; X] =A ^ dA =F ^F
4⇡ M =@X
4⇡ X
The fact that the Chern-Simons term is related to the 4-dimensional ✓ term was anticipated in (1.12) Written in this way, the Chern-Simons term is clearly gauge invariant
since it depends only on F and not A. Our worry, however, has transmuted to the question of whether it depends on the choice of 4-manifold X. How can we be sure that
we get the same answer if we chose a di↵erent 4-manifold X 0 which also has boundary@X 0 = M ? The di↵erence between the two answers involves the integral over the compact manifold Y = X [ X 0 , formed by gluing together X and X 0 along their common
boundary,
Z
k
0
F ^F
SCS [A; X] SCS [A; X ] =4⇡ Y– 405 –

We’re safe provided that this di↵erence is is 2⇡ times an integer, since then the partition
function, which depends on eiSCS , is independent on the choice of X. Clearly thisrequires
Z
1
F
F
^
2Z
(8.28)
2 Y 2⇡ 2⇡
So is this true? Well, actually no. Or, at least, not always! It turns out that (8.28) istrue only if the 4-manifold Y admits spinors or, more precisely, admits a mathematicalobject called a spin structure which tells you whether or not a fermion picks up aminus sign when it is transported around a loop. Any manifold that admits such a spin
structure is called a spin manifold. And (8.28) holds whenever Y is a spin manifold.
For example, Y = T4 , Y = S2 ⇥ S2 and Y = S4 are all spin manifolds. In thesecases (8.28) holds. To give you some sense of how this works, suppose that we takeY = S2 ⇥ S2 . Dirac quantisation means that the flux through each of the spheres must
be a multiple of 2⇡. If we take F = F1 + F2 , with Fn giving flux through the nth2-sphere, then
Z
Z
Z
1
F
F
^
=
F1
F2 2 Z2 S2 ⇥S2 2⇡ 2⇡
S2
S2
with the factor of 2 coming from the cross-term.
However, there are 4-manifolds Y which do not admit a spin structure. The simplest
R
example is Y = CP2 . In this case, Y (F/2⇡)^(F/2⇡) is an integer, not an even integer.

The upshot of this is that the Chern-Simons level k for a U (1) gauge group can be
integer valued provided that the theory admits fermions. But, otherwise, must be aneven integer. The simple “integrate by parts to get an extra factor of 2” prescriptionthat we used to get (8.27) sweeps all of these subtleties under the rug.
8.4.2 A Topological Phase of Matter
So what is the physics of Chern-Simons theory? Despite the simplicity of the action,
the physics is remarkably subtle. Let’s start with the basics. We’ll take the d = 2 + 1
dimensional gauge field to be governed by
Z
1
k µ⌫⇢
S = SMaxwell + SCS = d3 x
Fµ⌫ F µ⌫ +✏ Aµ @⌫ A⇢24e
4⇡
We can start by gaining some intuition from the classical equation of motion,@µ F µ⌫ +ke2 ⌫⇢✏ F⇢ = 04⇡– 406 –

(8.29)

In terms of the electric field Ei = F0i and the magnetic field B = F12 , Gauss’ law
becomes@i Ei =

ke2
B2⇡

(8.30)

which tells us that a magnetic field acts as a source for the electric field. This simple observation will underlie much of the physics of Section 8.6 where we discuss bosonizationin 3d.
What are the propagating excitations of the equations of motion (8.29)? Taking one
further derivative of the equations of motion, we can decouple electric and magnetic
fields to show that each component obeys the massive wave equation,✓ 2 ◆2✓ 2 ◆2
ke
ke
2
2
@ Ei
Ei = @ B
B=02⇡
2⇡(To do this, it’s perhaps simplest to first define the field Gµ = ✏µ⌫⇢ F⌫⇢ and show thatGµ obeys the massive wave equation.) We see that, at least classically, the excitations
do not propagate at the speed of light. Instead, they are exponentially damped. In thequantum theory, which means that we have a theory of massive excitations. The massof the photon is
mCS =

ke2
2⇡

Yet again, we find ourselves in a situation with a massive gauge boson. How should wethink of this phase?
We’ve already met other situations in d = 2 + 1 dimensions where the photon gets a
mass. There is the confining phase, driven by instantons, that we saw in Section 8.3,
in which the Wilson loop has an area law. And there is, of course, the Higgs phasein which a charged scalar field condenses and the Wilson line has a perimeter law. It
turns out that the Chern-Simons phase di↵ers from both of these. Instead, it is a novel
phase of matter, referred to as a topological phase.
Topological phases of matter are subtle. They typically have interesting things going on at energies E ⌧ mCS way below the gap, even though there are no physicalexcitations beyond the vacuum. We’ll explain below what these interesting things are.
Chern-Simons Terms are TopologicalBefore we address the novel physics of Chern-Simons theory, we first point out animportant property of the Chern-Simons action (8.21): it doesn’t depend on the metric– 407 –

of the background spacetime manifold. It depends only on the topology of the manifold.
To see this, let’s first look at the Maxwell action for comparison. If we were to couple
this to a background metric gµ⌫ , the action becomesZp
1 µ⇢ ⌫
SMaxwell = d3 xgg g Fµ⌫ F⇢
4e2
We see that the metric plays two roles: first, it is needed to raise the indices whenpcontracting fµ⌫ f µ⌫ ; second it provides a measureg (the volume form) which allows
us to integrate in a di↵eomorphism invariant way. Recall from our first lectures on
Quantum Field Theory that this allows us to quickly construct the stress-tensor of thetheory by di↵erentiating with respect to the metric,2
T µ⌫ = p@Lg @gµ⌫

In contrast, we have no need to introduce a metric when generalising (8.21) to curved
spacetime. This is best stated in the language of di↵erential geometry: A ^ dA is a
3-form, and we can quite happily integrate this over any three-dimensional manifold
Z
k
SCS =
A ^ dA
4⇡
This means that pure Chern-Simons theory knows nothing length scales. In particular,
the Wilson loop can exhibit neither area nor perimeter law, since both of these arestatements about lengths. Moreover, pure Chern-Simons theory has vanishing stresstensor.
Chern-Simons Theory on a Torus
If Chern-Simons theory has vanishing stress tensor, and no physical excitations, thenwhat can it possibly do? The answer is that the theory responds to low-energy probes
in interesting ways.
Here is a simple, yet dramatic way to probe the theory. We will place it on a spatial 2dimensional manifold ⌃. As we have seen, Chern-Simons theory knows nothing about
the metric on ⌃. However, as we now show, it does know about the topology andresponds accordingly.
For pure Chern-Simons theory (or, equivalently, the e2 ! 1 limit of Maxwell-ChernSimons theory), Gauss’ law (8.30) becomes
F12 = 0– 408 –

γ

γ

Figure 54:

Figure 55:

2

1

Although this equation is very simple, it can still have interesting solutions if the
background has some non-trivial topology. These are called, for obvious reason, flat
connections. It’s simple to see that such solutions exist on the torus ⌃ = T2 , where
one example is to simply set each Ai to be constant. Our first task is to find a gaugeinvariant way to parameterise this space of solutions.
We’ll denote the radii of the two circles of the torus T2 = S1 ⇥ S1 as R1 and R2 .
We’ll denote two corresponding non-contractible curves shown in the figure as 1 and
2 . The simplest way to build a gauge invariant object from a gauge connection is tointegratewi =I

dxj Aji

This is invariant under most gauge transformations, but not those that wind aroundthe circle. By the same kind of arguments that led us to (8.26), we can always constructgauge transformations which shift Aj ! Aj + 1/Rj , and hence wi ! wi + 2⇡. The
correct gauge invariant objects to parameterise the solutions are therefore the Wilson
loops
✓ I
◆
j
Wi = exp i
Aj dx = eiwi
iBecause the Chern-Simons theory is first order in time derivatives, these Wilson loops
are really parameterising the phase space of solutions, rather than the configuration
space. Moreover, because the Wilson loops are complex numbers of unit modulus,
the phase space is compact. On general grounds, we expect that when we quantise acompact phase space, we get a finite-dimensional Hilbert space. (We met an example ofthis in Section 2.1.3 when first describing Wilson lines.) Our next task is to understandhow to quantise the space of flat connections.

– 409 –

The canonical commutation relations can be read o↵ from the Chern-Simons action(8.21)[A1 (x), A2 (x0 )] =

i
k

2

(x

x0 )

)

[w1 , w2 ] =2⇡ik

The algebraic relation obeyed by the Wilson loops then follows from the usual BakerCampbell-Hausdor↵ formula,eiw1 eiw2 = e[w1 ,w2 ]/2 ei(w1 +w2 )which tells us that
W1 W2 = e2⇡i/k W2 W1(8.31)

But such an algebra of operators can’t be realised on a single vacuum state. This immediately tells us that the ground state must be degenerate. The smallest representationof (8.31) has dimension k, with the action
W1 |ni = e2⇡ni/k |ni and W2 |ni = |n + 1i
We have seen that on a torus ⌃ = T2 , an Abelian Chern-Simons theory has k degenerate
ground states. The generalisation of this argument to a genus-g Riemann surface tells
us that the ground state must have degeneracy k g . Notice that we don’t have to sayanything about the shape or sizes of these manifolds. The number of ground states
depends only on the topology. This is an example of topological order.
8.4.3 Non-Abelian Chern-Simons Theories
We’ve not had much to say about non-Abelian gauge theories in low dimensions. This
is not because they’re boring, but simply because there is enough to keep us busy
elsewhere. Here we make an exception and give a brief description of non-Abelian
Chern-Simons theory.
Like Yang-Mills, Chern-Simons is based on a Lie algebra valued gauge connection
Aµ . The non-Abelian Chern-Simons action is
✓
◆
Z
k
2i
3
µ⌫⇢
SCS =
d x ✏ tr Aµ @⌫ A⇢
Aµ A⌫ A⇢
(8.32)
4⇡
3
We’ve met this term before: the theta term in d = 3 + 1 dimensions can be written asa derivative of the Chern-Simons term (2.24). (It also arose in the same context when
discussing canonical quantisation of Yang-Mills (2.35).) Chern-Simons theories withgauge group G and level k are sometimes denoted as Gk .

– 410 –

Once again, we will find that the level must be integer, k 2 Z. This time, however,the computation is more direct than in the Abelian case. Under a gauge transformation,
we haveAµ ! ⌦ 1 Aµ ⌦ + i⌦ 1 @µ ⌦with ⌦ 2 G. The field strength transforms as Fµ⌫ ! ⌦ 1 Fµ⌫ ⌦. A simple calculationshows that the Chern-Simons action changes as
⇢
Z
k
1
3
SCS ! SCS +d x ✏µ⌫⇢ @⌫ tr (@µ ⌦ ⌦ 1 a⇢ ) + ✏µ⌫⇢ tr (⌦ 1 @µ ⌦)(⌦ 1 @⌫ ⌦)(⌦ 1 @⇢ ⌦)4⇡
3
The first term is a total derivative. The same kind of term arose in Abelian ChernSimons theories. However, the second term is novel to non-Abelian gauge theories,
and this is where the quantisation requirement now comes from. In fact, we have seen
this calculation before in Section 2.2.2 when discussing the theta angle in d = 3 + 1
Yang-Mills. On a spacetime manifold S3 (or on R3 with the requirement that gaugetransformations asymptote to the same value at infinity), gauge transformations arecharacterised by the homotopy group ⇧3 (SU (N )) ⇠= Z. The winding is counted by thefunctionZ1n(⌦) =
d3 S ✏µ⌫⇢ tr (⌦ 1 @µ ⌦ ⌦ 1 @⌫ ⌦ ⌦ 1 @⇢ ⌦) 2 Z(8.33)
2
24⇡ S3We recognise this as the final term that appears in the variation of the Chern-Simons
action. This means that the Chern-Simons action is not invariant under these large
gauge transformations; it changes as
SCS ! SCS +k24⇡ 2 n(⌦) = SCS + 2⇡k n(⌦)
12⇡

Insisting that the path integral, with its weighing eiSCS is gauge invariant then gives us
immediately our quantisation condition k 2 Z.
Wilson Loops
We have so far avoided talking about Wilson lines in Chern-Simons theories. There israther a lot to say. We will not describe this in detail here, but just sketch the key
idea.
In d = 3 Euclidean spacetime dimensions, a Wilson loop can get tangled. Mathematicians call closed curves in three dimensions knots, and there has been a great dealof e↵ort in trying to classify the ways in which they can get tangled. It turns out that

– 411 –

Chern-Simons theories provide one of the most powerful tools. For a given knot C,
we can compute the Wilson loop hW [C]i. In Chern-Simons theory the Wilson loop
exhibits neither an area law, nor a perimeter law. Instead, it depends on the detailsof the topology of the knot C. For each gauge group G, the Wilson loop gives a topological invariant which is a polynomial (roughly in q = e2⇡i/k .) In simple cases, these
topological invariants coincide with ones already understood by mathematicians (suchas the Jones polynomial), but they also o↵er a large number of generalisations. Edward
Witten was awarded the Fields medal, in large part for understanding this connection.
8.5 Fermions and Chern-Simons Terms
There is an intricate interplay between fermions in d = 2 + 1 dimensions and ChernSimons terms.
In signature ⌘ µ⌫ = diag(+1, 1, 1), the Cli↵ord algebra { µ ,by the 2 ⇥ 2 gamma matrices,
0

2

=

,

1

=i 1

3

,

⌫

} = 2⌘ µ⌫ is satisfied=i 3

The Dirac spinor is then a two-component complex object. In odd spacetime dimensions, there is no “ 5 ” matrix and, correspondingly, no Weyl fermions. In d = 2 + 1, we
can take the gamma matrices as above to be purely imaginary, which means that we canhave Majorana fermions. However, we won’t have a need for this real representation inwhat follows.
It will prove useful to understand the action of parity on fermions. As we saw in(8.22), in three dimensions parity acts as
x0 ! x0

,

x1 !

x1

,

x2 ! x 2

The Dirac action is then invariant if we take parity to act as
P:

7!

1

(8.34)

But this means that the fermion mass term necessarily breaks parity,
P: ¯ =† 07!

¯

where, to see this, you need to remember that ( 1 )† =1and ( 1 )2 =1.

This is di↵erent from what happens in d = 3 + 1 dimensions or, indeed, in any
even spacetime dimension. There parity flips the sign of all spatial dimensions and,correspondingly, the Dirac action is invariant if we take P : 7! 0 . This means thatin even spacetime dimensions, ¯ is even under parity; in odd spacetime dimensions¯ is odd.

– 412 –

We can understand why this is by counting degrees of freedom. In d = 3 + 1
dimensions, the Dirac spinor has 4 components. When we quantise a massive fermion,
we get two particle states – spin up and spin down – and the same anti-particle states.
But a Dirac fermion in d = 2 + 1 dimensions has only two components, and so we musthave half the number of particle states of the d = 3 + 1 theory. The pair that we keep
is dictated by the sign of the mass, and by CPT invariance: if we have a particle withspin, or angular momentum, + 12 , the theory must also include an anti-particle of spin
1
. But this necessarily breaks parity: the theory has a particle of spin + 12 but no2
particle of spin 12 .
8.5.1 Integrating out Massive Fermions
Let us take a single Dirac fermion, of mass m, coupled to a U (1) gauge field Aµ . The
action isZ/ +m¯
S = d3 x i ¯D
If we care about physics at energies below the fermion mass m, we can integrate out
the fermion. We work in Euclidean space. The fermion then gives a contribution to
the low-energy e↵ective action for the gauge field,/ + m = Tr log i @/+Se↵ = log det iDµ

Aµ + mWe expand this as,/ + m + Tr log
Se↵ = Tr log i @

✓

1
µ
Aµ
/ +m
i@

◆

1
+ Tr log
2

✓

1
1
µ
⌫
Aµ
A⌫
/ +m
/ +m
i@
i@

◆

The first term is an overall constant, and the second term cannot lead to anything gauge
invariant. But the third term holds something interesting. If we give the backgroundfield Aµ momentum p, then the trace over momenta corresponds to the diagram,
✓
◆
Z
1
d3 k
1
1
µ
⌫
= Aµ ( p)A⌫ (p)
tr
2
(2⇡)3
(/p + k/ ) + m k/ + m
✓
◆
Z
3
1
dk
p/ + k/ m µ k/ m ⌫= Aµ ( p)A⌫ (p)
tr
2
(2⇡)3
(p + k)2 + m2 k 2 + m2
where we’ve used the fact that, after the Wick rotation, each gamma matrix squares
to 1.

– 413 –

+ ...

The trace picks out the non-vanishing gamma matrix structure. There will be acontribution to the Maxwell term; that doesn’t interest us here. Instead, we careabout the term we get when three gamma matrices are multiplied together. The trace
structure gives
tr

⇢ µ ⌫

2✏µ⌫⇢

=

The resulting term is=✏µ⌫⇢

Aµ ( p)A⌫ (p) p⇢

Z

d3 k
m
3
2
(2⇡) ((p + k) + m2 )(k 2 + m2 )We’re interested in this integral in the infra-red limit, p ! 0, where it is given by
Z
Z 1
d3 k
m
1
mk 2
1
=
dk
=
3
2
2
2
2
2
2
2
(2⇡) (k + m )2⇡ 0(k + m )8⇡|m|Putting this together, the 1-loop diagram gives
lim

p!0

=

1 m µ⌫⇢✏ Aµ ( p)A⌫ (p) p⇢
8⇡ |m|

Back in real space, this gives us the leading term to the low energy e↵ective action
Z
i sign(m)Se↵ =d3 x ✏µ⌫⇢ Aµ @⌫ A⇢(8.35)
4⇡
2
There are a number of interesting things to point out about this result. First, thee↵ective action comes with a power of i; this is expected for the Chern-Simons term in
Euclidean space, and follows from Wick rotating terms with an ✏ symbol.
Second, and more surprisingly, the fermion does not decouple in the limit m ! 1.
After integrating out a massive field, one typically generates terms in the e↵ective actionthat scale as a power of 1/m. Not so for the Chern-Simons term: it is proportional
to the sign of the mass. This behaviour holds for fermions in any odd spacetime
dimensions; we met a similar example in d = 4 + 1 when discussing anomaly inflow in
Section 4.4.2.
Finally, and most importantly, the e↵ective action (8.35) is not gauge invariant! It
is a Chern-Simons term (8.21) with level k = ± 12 . Yet, we saw in the previous section,that the Chern-Simons term is only gauge invariant for k 2 Z. With k = ± 12 , the signof the partition function can flip under gauge transformations.

– 414 –

What are we to make of this? It appears that a single massive Dirac fermion, coupled
to a U (1) gauge field, is inconsistent. This is very much reminiscent of the gauge
anomalies that we met in d = 3 + 1 dimensions in Section 3. However, we shouldn’t be
too hasty. After all, anomalies in d = 3 + 1 dimensions were strictly related to massless
fermions, and here we’re dealing with a massive fermion. What’s going on?
Indeed, we were sloppy in how we deal with UV divergences in the calculation above.
They do not arise in the calculation of the Chern-Simons term, but they will surely be
important if we compute other quantities and, as in any quantum field theory, we needa way to regulate them. To achieve this, we introduce a Pauli-Villars regulator field,
together with suitable counterterms. We take the Pauli-Villars field to have real mass
⇤U V > 0. The regulated Dirac determinant is then
/ + m)
det(iD/ + ⇤U V )det(iD
This gives two contributions to the Chern-Simons term; one from our fermion, and one
from the regulator. The e↵ective action for the gauge field then becomes✓◆Z/ + m)
det(iD1
1=
sign(m)d3 x ✏µ⌫⇢ Aµ @⌫ A⇢/ + ⇤U V )2⇡2det(iD

which vanishes when m > 0 but gives a Chern-Simons term of level k = 1 when m < 0.
In other words, when the regulated fermion determinant is defined more carefully, there
is no problem with gauge invariance.
The resulting situation is notationally inconvenient. Usually we would like to write
down an action as shorthand for a quantum field theory, even though we know that to
fully define the theory really requires a statement about how we regulate. The issue
above means that the sign of the mass of the Pauli-Villars regulator matters in a crucial
fashion. To avoid this, we are often sloppy and pretend that we’ve already integrated
out the Pauli-Villars field to generate a bare Chern-Simons term with level k = 12 in
the action.
More generally, we can couple Nf Dirac fermions to a U (1) gauge field with theleading terms in the action given by
Nf
Z
X
1
k µ⌫⇢
3
µ⌫
/ i + mi ¯i i
S= dx
Fµ⌫ F +✏ Aµ @⌫ A⇢ +i ¯i D4e2
4⇡i=1

Using the convention that the Chern-Simons term already includes the contributions
from Pauli-Villars fields, gauge invariance requires
Nf
k+
2Z
2

– 415 –

This interplay between the level k and the number of fermions is sometimes referred toas the parity anomaly. It’s not a great name since the theory with fermion masses is
not parity invariant to begin with.
8.5.2 Massless Fermions and the Parity Anomaly
We can gain a slightly di↵erent perspective on the ideas above by considering a massless
fermion coupled to a U (1) gauge field, Aµ . The action is now
S=Z/
d3 x i ¯D

The transformation (8.34) ensures that the classical action is invariant under parity,
provided that we also act with A1 ! A1 .
The classical action is invariant under parity. But what about the partition function.
To answer this, we must make sense of the determinant of the Dirac operator,/
Z[A] = det iD
As above, we work in Euclidean space. The Dirac operator is Hermitian, which meansthat it has real eigenvalues,
/ n=
iD

n 2 R

n n

So formally we can write
Z=

Y

n

n

Of course, this formula is divergent and so we must work to make sense of it. For/ Roughly
now, we would like to ask the following question: what is the sign of det(iD).
speaking, this must be the di↵erence between the number of negative eigenvalues andthe number of positive eigenvalues. But, as there are an infinite number of each, it is
not clear how to count them.
Why do we care so much about the sign? The problem comes if we try to reconcile agiven sign with the requirements of gauge invariance. Suppose that we start with some/ has a specific sign. Then it better begauge configuration A?µ and decide that det(iD)!
the case that, for any gauge configuration Aµ , related to A?µ by a gauge transformation,/ remain the same.
the sign of det(iD)– 416 –

At this point, the discussion may be ringing bells. It is entirely analogous to theSU (2) anomaly that we described in Section 3.4.3. We proceed in a very similar way.
Consider the 1-parameter family of gauge configurations,
Aµ (s; x) = (1s)A?µ (x) + sA!µ (x)(8.36)

This has the property that it interpolates from A?µ when s = 0 to A!µ when s = 1.
The question that we would like to answer is: how many eigenvalues pass throughzero and change sign as we vary s 2 [0, 1]. To answer this, we can consider the gauge
configuration Aµ (s; x) in (8.36) to live on the four manifold I ⇥ R3 , where I is theinterval parameterised by 0  s  1.
The number of times that the an eigenvalue crosses zero is given by the index of
the Dirac operator. This is the object that we introduced in Section 3.3.1 where, on aclosed four manifold, the Atiyah-Singer index theorem allowed us to writeZ1/ 4d ) =Index(iD
d4 x ✏µ⌫⇢ Fµ⌫ F⇢
32⇡ 2 I⇥R3
In 4d, the index counts the di↵erence between the number of left-handed and righthanded zero modes. For our purposes, it tells us the di↵erence between the number of
eigenvalues that switch from positive to negative, and those which switch from negative
to positive. In other words, under the gauge transformation A0µ ! A!µ , the partition
function of the massless fermion changes as
Z ! Z ( 1)Index(iD/4d )There is no reason for this index to be even. We see, once again, that without regularisation the sign of the partition function can change under a suitable gauge transformation.
What happens if we now include a regulator? In mathematics, a suitably regulated/ is known as the Atiyah-Patodi-Singer etasum of the signs of the eigenvalues of iD
invariant. It is defined by⌘(A) = lim+
✏!0

X

2

e ✏ n sign( n )nWe then define a regulated version of the fermion partition function as/ e i⇡⌘(A)/2
Z = | det(iD)|– 417 –

The ⌘ invariant depends on the background gauge field A. The Atiyah-Patodi-Singer
index theorem provides an expression for ⌘ in terms of the gauge field. If we restrict tothe generic situation where the gauge field has no zero modes, then one can show thatZ
1
⇡ ⌘(A) =d3 x ✏µ⌫⇢ Aµ @⌫ A⇢
4⇡
This reproduces the expression that we found previously from the Pauli-Villars regularisation. In general, the eta-invariant is the more mathematically rigorous way todescribe what’s happening as it allows one to track what happens as eigenvalues passthrough zero.

8.6 3d Bosonization
In two spacetime dimensions, there is not much of a distinction between bosons andfermions. The map between them is known as bosonization and was described in
Section 7.5.
In three spacetime dimensions, bosons are not the same as fermions. We can tellwhich one we have in the same way as we would in four dimensions. Given a pair ofparticles we can rotate them by 180 , keeping them well separated. The wavefunction
for a pair of bosons will come back to itself, while the wavefunction for a pair of fermionscomes back with a minus sign.
Nonetheless, it is possible to use Chern-Simons terms to change statistics of an
excitation from a boson to a fermion. This process is referred to as 3d bosonization.
8.6.1 Flux Attachment
To get a feel for what’s going on, it’s useful to first revert to some non-relativistic
physics. Consider Chern-Simons theory coupled to a current J µZk µ⌫⇢
S = d3 x✏ Aµ @⌫ A⇢ + Aµ J µ(8.37)
4⇡
We can insert a test particle of unit charge by taking J µ = 2 (x). How does thegauge field respond? Gauss’ law tells us that the charged particle is accompanied by afractional magnetic flux,
1
1
B=
2⇡
k

2

This is referred to as flux attachment.

– 418 –

(x)(8.38)

Now consider two such particles. We will exchange them to determine their quantum
statistics. The wavefunction will pick up a factor of ±1 depending on whether theoriginal particles were fermions. However, there is a second contribution to the phaseof the wavefunction that comes from the Aharonov-Bohm e↵ect.
Recall that a particle of charge q moving around a flux
picks up a phase eiq .
But because of flux attachment (8.38), the particles carry both charge q = 1 and flux
= 2⇡/k. If we move one particle all the way around another, we will get a phase eiq .
But the statistical phase is defined by exchanging particles, which consists of only half
an orbit (followed by a translation which contributes no phase). So, after exchange,
the expected statistical phase is± eiq /2 = ±ei⇡/kwhere we take the + sign if our original particles are bosons and thesign if they
were fermions. We see that the e↵ect of the Chern-Simons term is to transmute thequantum statistics of the particles. In particular, if we take a Chern-Simons term atlevel k = ±1, what were bosons become fermions and vice versa. Once again, we seethat the topological nature of the Chern-Simons term endows it with seemingly magic
infra-red properties: it can change the behaviour of far separated particles, even though
it has no propagating degrees of freedom.
For |k| > 1, the particles are neither bosons nor fermions. Instead they carry fractional quantum statistics. Such particles are called anyons and are allowed only in
d = 2 + 1 dimensions. You can read more about them in the lecture notes on the
Quantum Hall E↵ect where they play a prominent role.
A Famously Fiddly Factor of 2
The calculation above contains an annoying factor of 2 that we’ve swept under therug. Here’s the issue. As the charge q in the first particle moved around the fluxin the second, we picked up a phase eiq . But you might think that the flux of thefirst particle also moved around the charge q of the second. So surely this should give
another factor of eiq . Right? Well, no. To see why, it’s best to just do the calculation.
For generality, let’s take N particles sitting at positions xa (t) which, as the notationshows, we allow to change with time. The charge density and currents are
0

J (x, t) =

N
X

2

(x

xa (t)) and J(x, t) =

a=1

N
X
a=1

– 419 –ẋa 2 (xxa (t))

The equation of motion from (8.37) is11Fµ⌫ = ✏µ⌫⇢ J ⇢2⇡kand can be easily solved even in this general case. We work in Coulomb gauge with
A0 = 0 and r · A = 0. The solution is thenN

1 X ij xj xja (t)Ai (x, t) =✏k a=1
|x xa (t)|2(8.39)

This follows from the standard methods that we know from our Electromagnetism
lectures, but this time using the Green’s function for the Laplacian in two dimensions:r2 log |x y| = 2⇡ 2 (x y). This solution is again the statement that each particle
carries flux 1/k. However, we can also use this solution directly to compute the phasechange when one particle – say, the first one – is transported along a curve C. It issimply✓ I◆
exp i A · dx1
C

If the curve C encloses one other particle, the resulting phase change can be computed
to be e2⇡i/m . As before, if we exchange two particles, we get half this phase, or ei⇡/k .
This, of course, is the same result we got above.
8.6.2 A Bosonization Duality
The discussion above shows that Chern-Simons terms can turn bosons into fermions andvice-versa. However, it holds only for massive particles, and cannot be easily generalised
to massless particles, let alone to relativistic quantum field theories. Nonetheless, it
is suggestive that it may be possible to write down a quantum field theory of bosons
coupled to Chern-Simons terms that has a dual interpretation in terms of fermions. As
we now explain, it is thought that this is indeed the case.
Before we proceed, we’re going to make a small change in notation. In what follows,
there will be lots of U (1) gauge fields floating around. Some of them will be dynamical,
while others will be background gauge fields that we couple to currents. To distinguishbetween these, we use the following convention: dynamical gauge fields will be written
in lower case, e.g. aµ . Meanwhile, background gauge fields will be written in upper
case, e.g. Aµ .

– 420 –

This convention di↵ers from what we’ve used throughout these lectures, where wetypically refer to all gauge fields, dynamical or background, as Aµ . It is, however,a standard convention in condensed matter physics where the true electromagnetic
gauge field Aµ is typically a background field, describing electric or magnetic fields
that the experimenter has chosen to turn on. In contrast, 3d dynamical gauge fields
aµ are always emergent excitations, arising from some collective behaviour of strongly
coupled electrons.
Consider the following theory, that we refer to as Theory A: a complex scalar field
coupled to a U (1) gauge field, with Chern-Simons term at level k = 1,
Z
1
1 µ⌫⇢
SA [ , a] = d3 x
fµ⌫ f µ⌫ +✏ aµ @⌫ a⇢ + |Dµ |2 m2 | |2
| |4 (8.40)
2
4e
4⇡
2
This is the Abelian Higgs model (8.12), but with the addition of a Chern-Simons term.
Just as before, it is straightforward to analyse in the limits m2
e2 and m2 ⌧ e2
where it is a theory of weakly interacting massive particles. But we’d like to understandwhat happens in the strongly coupled regime. We will argue below that as we vary them2 from positive to negative, there is a unique second order phase transition, roughly
at m = 0. You can think of this gapless theory as the XY critical point, coupled
to a Chern-Simons gauge field U (1)1 . Below, we will conjecture an alternative, andsomewhat simpler, description.
In the infra-red limit e2 ! 1, the Gauss’ law constraint gives rise to the local flux
attachment condition,f12+ ⇢scalar = 02⇡(8.41)

where ⇢scalar is the charge density of the scalar field . In the non-relativistic setting –which can be invoked when m2
e2 – we viewed this as attaching flux to every scalar
excitation and saw that, for k = 1, this turns a boson into a fermion. In the relativistic
setting, it turns out to be more appropriate to think of attaching a scalar to every flux.
To see this, first note that the theory has a conserved global symmetry, with the
topological current (8.3)
µ
jtop
=

1 µ⌫⇢
✏ @ ⌫ a⇢2⇡

(8.42)

We know from our earlier discussion in Section 8.1 that the local operators which carrycharge under this current are monopole operators M(x), which insert magnetic flux

– 421 –

at a point. The flux attachment (8.41) is telling us that, in the presence of a ChernSimons term, these monopole operators are not gauge invariant. We can make themgauge invariant only by dressing them with some scalar charge ⇢scalar . Schematically,
we refer to the gauge invariant composite operator as M .
How do we do this less schematically? The right way to proceed is to solve theequation of motion for the scalar in the presence of a Dirac monopole. We then treat
each mode quantum mechanically: the flux attachment condition (8.41) tells us that
we should excite a single mode. The monopole operator with the lowest dimension willcorrespond to exciting the lowest energy scalar mode.
We won’t go through this full calculation. However, the key physics can be seenfrom a simple calculation that we did back in Section 1.1: a charged particle moving
in a minimal Dirac monopole receives a shift of ~/2 to its angular momentum. (See,in particular, equation (1.9).) This means that exciting any bosonic mode will shift
the angular momentum of the monopole to become 1/2-integer. But, in a relativistic
theory, the spin-statistics relation must hold. If our gauge invariant monopole operator
M has spin 1/2, then it must also be a fermion.
We see that this argument leads to the same result as before: a bosonic theory coupled
to a U (1) Chern-Simons gauge field at level k = 1 is really a theory of fermions. The
obvious question is: what theory of fermions?
It is conjectured that, close to the critical point, the bosonic theory (8.40) is reallyjust a free Dirac fermion! In other words, it can be equivalently described asZ/
SB [ ] = d 3 x i ¯ @m0 ¯(8.43)The map is very similar to that of particle-vortex duality that we saw in Section 8.2.1.
In particular, the fermion is described by the dressed monopole operator in Theory A,
M!

while the U (1) currents map between themselves
µ
jtop
=

1 µ⌫⇢
✏ @ ⌫ a⇢2⇡!

jµ = ¯ µ(8.44)

Checking the Topological Phases
Let’s now look for some evidence that this claimed duality is correct. In the caseof particle-vortex duality, we checked that the theories looked similar in the weakly
coupled regimes |m2 |e2 . We can try to do something similar here.

– 422 –

This is simplest for Theory B. To study the relevant physics, we couple the current(8.44) to a background gauge field Aµ . The partition function for each theory thendepends on this background field. For Theory B it is✓◆ZZ
1 1 µ⌫⇢3
ZB [A] = D exp iSB [ ] + i d x jµ Aµ✏ A⌫ @⌫ A⇢
2 4⇡
Note that we are using the convention described in Section 8.5, in which the half-integer
Chern-Simons term arising from the Pauli-Villars regulator field is shown explicitly inthe action. We have chosen to add this term with level k = 1/2.
When the fermions are massive, m0 6= 0, we can integrate them out and generate ane↵ective theory for the background fields Aµ . The lowest dimension term is a ChernSimons interaction for Aµ ,!
k̃ µ⌫⇢
Z[A] = exp i ✏ Aµ @⌫ A⇢ + . . .
(8.45)
4⇡
From our discussion in Section 8.5, we know that after integrating out the massive
fermion the Chern-Simons level for the background gauge field will be(0m0 > 01k̃ = ( 1 + sign(m0 )) =2
1 m0 < 0It may seem odd to write down an action for background fields which don’t fluctuate,
but there’s important information in the coefficient k̃; it is the Hall conductivity of the
topological gapped phase. This follows by using the partition function Z[A] to computethe response of the current j µ to a background electric field
hj µ (x)i =ilog Z[A]Aµ (x))

hji i =k̃
✏ij Ei2⇡

You can read (a lot) more about the Hall conductivity in the lectures on the Quantum
Hall E↵ect.
We would like to see how this e↵ect is encoded in the bosonic Theory A. We couple
the background gauge field Aµ to the topological current (8.42) to get the partition
function
✓
◆
Z
1 µ⌫⇢
ZA [A] = D Da exp iSA [ , a] + i ✏ Aµ @⌫ a⇢2⇡– 423 –

where we’re neglecting gauge fixing terms. This time we only have a scalar field, which
does not shift the level of the Chern-Simons term when integrated out. Nonetheless,
we can still reproduce the result (8.46) for the Hall conductivity. To see how thisworks, let’s start with the mass m2
e2 where, at low energies, the scalar field simply
decouples, leaving us with the e↵ective action
Z
1 µ⌫⇢
1 µ⌫⇢
Se↵ [a, A] = d3 x
✏ aµ @ ⌫ a⇢ +✏ A µ @ ⌫ a⇢4⇡
2⇡
The equation of motion for the dynamical gauge field a is simply a =this back in, given the e↵ective action (8.45) with k̃ = 1.

A. Substituting

What happens when m2 ⌧ e2 ? In this case the scalar field condenses and the
dynamical gauge field a becomes gapped. This extra term kills the Hall conductivity,
leaving us with (8.45) with k̃ = 0. We see that the scalar field does reproduce thetopological phases of the the fermion theory as promised. This requires the map,
m2

!

m0

)

†

!

¯

The agreement between the topological phases is promising, but a long way from demonstrating the claimed duality between Theory A (8.40) and the free fermion (8.43). There
are a number of other routes which lead us to the duality (including large N methods,
holography, lattice constructions and supersymmetry) but we will not discuss them
here. Instead we will assume that bosonization duality holds and ask: what can we dowith it?
8.6.3 The Beginning of a Duality Web
We will now show how, starting from the bosonization duality, we can derive further
equivalences between quantum field theories. First, some conventions. We will revert
to form notation for the gauge fields, and write the Chern-Simons terms as
1 µ⌫⇢
1✏ aµ @ ⌫ a⇢ =
ada
4⇡
4⇡
1 µ⌫⇢
1
1
✏ A µ @ ⌫ a⇢ =
Ada =
adA
2⇡
2⇡
2⇡
Both of these are correctly normalised as explained in Section 8.4: they can be added
to the action only with integer-valued coefficients. We will denote the gauge field underwhich matter is charged by adding a subscript to the covariant derivative like this,Da = @– 424 –

ia

The spacetime index on the derivatives will be suppressed. In what follows, the distinction between dynamical gauge fields and background gauge fields will be crucial.
As we mentioned previously, they are distinguished by case. Lower case gauge fields,a, b, c, . . . will always be dynamical; upper case gauge fields A, B, C, . . . will always bebackground.
In this notation, we write the 3d bosonization duality that we described above as anequivalence between two theories|Db |2| |4 +

1
1
bdb +
Adb
4⇡
2⇡

!

/A
i ¯D

1 1
AdA
2 4⇡

(8.46)

Much of this expression is shorthand. First, we have set the mass terms to zero on
both sides. This really means that we tune to the critical point. On the fermionic side
this is obvious, but the scalar side includes a | |4 term which is taken to mean that weflow to the Wilson-Fisher fixed point of the theory, rather than the free fixed point. Of
course, we don’t literally get to the Wilson-Fisher by simply setting m2 = 0; instead
we must tune m2 , or more generally the coefficient of the relevant operator, as we flow
to the IR to hit the critical point. All of this is buried in the notation above.
Second, we reiterate that the scalar in the above expression is charged under adynamical gauge field, which we have called b to prepare us for some manipulations
ahead. This means that we integrate over (gauge equivalent) configurations of b in thepath integral. In contrast, the fermion is charged under the background field A. We
can read o↵ the duality map (8.44) between currents by seeing which terms on bothside are coupled to A. Finally, we’ve omitted nearly all the details of the regularisationof the field theory, with one exception: the level 1/2 Chern-Simons term on the righthand-side can be thought of as coming from integrating out a Pauli-Villars regulator.
This was explained in Section 8.5. (A warning: some places in the literature adopt adi↵erent convention where this level 1/2 Chern-Simons term remains hidden in theregulator.)
At this point we start to play with these two theories. Both sides of the duality (8.46)
have a background U (1) gauge field A. The key idea is to promote this to a dynamical
gauge field. This is misleadingly easy in our notation: we simply write a instead of A.
As we explained in Section 8.1, gauging a U (1) symmetry in d = 2 + 1 results in a new
global symmetry,
jµ =

1 µ⌫⇢
✏ @⌫ a⇢2⇡– 425 –

1
We couple this to a background gauge field C. This means that we add 2⇡
AdC to bothsides of (8.46), and then make A ! a dynamical. This results in a new duality,|Db |2| |4 +

1
1
1
bdb +
adb +
adC
4⇡
2⇡
2⇡

!

/a
i ¯D

1 1
1
ada +
adC
2 4⇡
2⇡

The number of gauge fields on the left-hand side are proliferating. But, at this point,something nice happens: the gauge field a only appears linearly in the action. Thismeans that it acts as a Lagrange multiplier, setting db = dC. But, this, in turn,freezes the first dynamical gauge field b to be equal, up to gauge connection, to the new
background field C. The upshot is that we end up with a scalar field theory with nodynamical gauge fields at all, and the duality|DC |2| |4 +

1
CdC
4⇡

!

/a
i ¯D

1 1
1
ada +
adC
2 4⇡
2⇡

(8.47)

This is a new equivalence between two, seemingly very di↵erent looking, theories. The
left-hand-side is something very familiar: it is the XY Wilson-Fisher fixed point. In
contrast, the right-hand side is the a strongly coupled U (1) gauge theory. The claim isthat these two fixed points are the same, so!

XY Wilson-Fisher

U (1) 1/2 coupled to a Dirac fermion

From our first bosonization duality, we have derived another. Similarly, we can go inreverse: starting from the equality of partition functions (8.47), it is not hard to derive
the original (8.46).
We can continue in this vein, adding di↵erent matter fields and gauging global symmetries, to derive an infinite number of dualities between di↵erent 3d Abelian theories
with Chern-Simons terms. This is referred to as the duality web. Below we give just a
handful of interesting examples.
8.6.4 Particle-Vortex Duality Revisited
Our second bosonization duality (8.47) includes a Chern-Simons coupling for the background field C on the left-hand-side. Since we don’t integrate over the backgroundfield, there is nothing to stop us taking this term onto the other side of the equation.
We will also take this opportunity to rename some of the variables. The duality (8.47)
is equivalent to|DA |2

| |4

!

/b
i ¯D

1 1
1
bdb +
bdA
2 4⇡
2⇡

– 426 –

1
AdA
4⇡

(8.48)

Having moved the background Chern-Simons term to the other side, we now play the1same game as before: we add a term 2⇡
AdC, and then again promote A to a dynamical
field, A ! a. We now have

1
1 1
1
1
1
/b
adC
!
i ¯D
bdb +
bda
ada +
adC
2⇡
2 4⇡
2⇡
4⇡
2⇡
Again, there’s a lot of gauge fields on the right-hand-side. Now a does not appear
linearly as a Lagrange multiplier, but quadratically. Still, it is begging to be integratedout by imposing the equation of motion a = b + C, leaving us with
|Da |2| |4 +

1
1 1
1
1
/b +
adC
!
i ¯D
bdb +
bdC +
CdC (8.49)
2⇡
2 4⇡
2⇡
4⇡
This is still a bosonization duality, relating a scalar theory to a fermionic theory. But
the right-hand-side is very nearly the same expression that we started with in (8.48),
but with one important di↵erence: two of the Chern-Simons have their sign flipped. In
fact, we we send C ! C, all of the Chern-Simons terms have their sign flipped. In
other words, this partition function describes the time reversal of the theory in (8.48).
|Da |2| |4 +

As we have seen, Chern-Simons terms break time reversal, so one would not naively
expect that U (1)1/2 coupled to a Dirac fermion is time reversal invariant. However, if
we take the time reversal of the duality (8.48), we have
1 1
1
1
bdb
bd( C) +CdC(8.50)
2 4⇡
2⇡
4⇡
By charge conjugation we can replace D C ! DC . The left-hand-side is once again
the XY critical point. It is clearly time-reversal invariant. The duality tells us that
U (1)1/2 coupled to a massless fermion must be secretly time reversal invariant: it must
emerge as a discrete symmetry of the quantum theory.
|D C |2

| |4

!

/b +
i ¯D

Combining (8.49) together with (8.50) gives us yet another duality. It is1
adC! |DC |2 | |42⇡
But this is precisely the statement of particle vortex duality that we discussed in Section
8.2.1: the left-hand-side is the Abelian Higgs model while the right-hand-side is the
XY model. We learn that particle-vortex duality = bosonization2 .
|Da |2| |4 +

8.6.5 Fermionic Particle-Vortex Duality
Above we have managed to use 3d bosonization to derive a duality between purely
bosonic theories. We might ask: can we do something similar to derive a duality
between purely fermionic theories? The answer is yes. But, there will be a new subtlety
that we have to address.

– 427 –

We can see this subtlety by retracing the steps above. To derive bosonic particlevortex duality, we started with the bosonization dual (8.47), moved the background
Chern-Simons term to the other side, and then promoted the background gauge field
to a dynamical one. To derive a fermionic particle-vortex duality, it is natural toattempt the same manoeuvres for our original bosonization duality (8.46),
1
1
1 1
/A
bdb +
Adb
! i ¯DAdA(8.51)
4⇡2⇡
2 4⇡
But we immediately run into a stumbling block: we can’t move the background ChernSimons term to the other side because it is half-integer valued. It is needed on theright-hand-side to ensure that the fermion partition function is gauge invariant.
|Db |2| |4 +

To get around this, we will stipulate that the background gauge field A only admits
flux quantised asZ1
dA 2 2Z
2⇡This is twice the usual requirement. We can then write
A = 2C
with C a background gauge field whose flux is correctly quantised. The duality (8.51)
is then1
2
2/ 2C|Db |2 | |4 +
bdb +
Cdb
! i ¯D
CdC(8.52)
4⇡
2⇡
4⇡
All Chern-Simons terms are now properly quantised. But the fermion on the righthand-side has charge 2 under the gauge field C. If we give a fermion of charge q a mass
m and integrate it out, it will generate a Chern-Simons term with level 12 q 2 sign(m).
(This follows from the fact that the one-loop diagram in Section 8.5 has two insertionsof the photon-fermion vertex.) So integrating out a fermion of charge 2 generates aninteger-valued Chern-Simons level and there is no problem with the parity anomaly.
Now let us play games with this theory. We will move the CdC background Chern1
Simons term to the other side, add 2⇡
BdC to both sides, and finally breath life into C
to make it dynamical, C ! a. We have
1
2
2
1
1
/ 2a +
bdb +
adb +
ada +
adB
! i ¯D
adB
4⇡
2⇡
4⇡
2⇡
2⇡
The mess of mixed Chern-Simons terms on the left-hand-side is easily dealt with: we
simply define the new linear combination|Db |2| |4 +â = a + b– 428 –

Then we find
1
1
2
1
1
/ 2a +
bdb
bdB +âdâ +âdB
! i ¯D
adB
4⇡
2⇡
4⇡
2⇡
2⇡
But the first four terms in this expression – those which involve and b — coincide withthe time-reversal of the left-hand-side of (8.51). We can then use the duality (8.51) toreplace them, leaving us with the promised fermion-fermion duality,|Db |2

| |4

1 1
2
1
1
/ 2a +
AdA +âdâ +âdA
! i ¯D
adA
2 4⇡
4⇡
2⇡
2⇡
where we’ve taken this opportunity to rename the background field A.
/A +
i ¯D

What is this final expression telling us? The right-hand-side is a U (1) gauge theory
coupled to a single Dirac fermion of charge 2. The left-hand-side is very almost a freefermion. But it also includes a decoupled topological theory, U (1)2 , described by the
dynamical gauge field â. We learn that
U (1) with Dirac fermion of charge 2!

Free Dirac fermion + U (1)2This is the fermionic version of particle-vortex duality, with the monopole operatorsof the gauge theory identified with the fermion. A closely related duality was firstsuggested by Son in the context of the half-filled Landau level. It has also been invokedin the context of topological insulators.
8.7 Further Reading
Quantum field theories in d = 2 + 1 dimensions have a rather special relation to thereal world because, after a Wick rotation, many of them (but not all of them!) can beviewed as statistical field theories in d = 3 + 0 dimensions, where they describe systems
near critical points. For example, 4 scalar field theory in d = 3 dimensions describes
the water boiling in your kettle. (Admittedly, you might need to put a fairly tight lid
on the kettle.)
From the high energy perspective, d = 2 + 1 dimensions o↵er another arena to study
questions about gauge theories that seemed too challenging in d = 3 + 1. Polyakov’s
demonstration of confinement [159, 160], driven by the proliferation of instantons(monopoles), was a highlight in this regard. Similarly, particle vortex duality was
first introduced by Peskin [152], in an attempt to see whether a similar duality in
d = 3 + 1 could help explain confinement. This was subsequently rediscovered in thecondensed matter community by Dasgupta and Halperin, who also performed numerics
to find convincing evidence of a second order phase transition [37]. Both of these papers
originally expressed the duality in terms of lattice theories; the continuum version that
we described here was first proposed in [61].

– 429 –

Chern-Simons theory was introduced by Deser, Jackiw and Templeton [42, 43], initially as a surprising, gauge invariant mechanism to give the three dimensional photon
a mass. The depth of the theory became apparent with Witten’s Fields medal winningwork on knot invariants [228], and the connection to WZW models [53]. The interplay between massive fermions and Chern-Simons terms was discovered in [149] and[168, 169]; a more modern perspective was provided by Witten in [230]. A very clear
discussion of the properties of Chern-Simons theories can be found in the lectures byDunne [49]. You can read more about the subtleties related to the quantisation of
Abelian Chern-Simons theories in the appendices of [175] and [176]The story of 3d bosonization has a long and complicated history. The idea that one
can use Chern-Simons terms to transmute the statistics of non-relativistic particles frombosons to fermions was pointed out by Wilczek and Zee [211]. Polyakov was the first to
conjecture that there might be a relativistic version of bosonization, but he missed theneed to bosonize at the Wilson-Fisher fixed point [161]. The full story came by bringing
together a wonderfully diverse set of ideas from both high energy and condensed matterphysics. These include dualities in supersymmetric theories [110], large N bosonizationand its relation to holography [76, 5, 6, 7], and physics associated to superfluids [13], thehalf-filled Landau level [186] and topological insulators [200, 136]. The web of dualities
among Abelian gauge theories, relating bosonization and particle-vortex duality, wasfirst described [119, 176].

– 430 –

References[1] L. F. Abbott, “Introduction to the Background Field Method,” Acta Phys. Polon. B 13,
33 (1982).
[2] S. Adler, “Axial Vector Vertex in Spinor Electrodynamics”, Phys. Rev. 177, 2426 (1969)[3] S. L. Adler and W. A. Bardeen, “Absence of higher order corrections in the anomalous
axial vector divergence equation,” Phys. Rev. 182, 1517 (1969).
[4] O. Aharony, N. Seiberg and Y. Tachikawa, “Reading between the lines offour-dimensional gauge theories,” JHEP 1308, 115 (2013) [arXiv:1305.0318 [hep-th]].
[5] O. Aharony, G. Gur-Ari and R. Yacoby, “d=3 Bosonic Vector Models Coupled to
Chern-Simons Gauge Theories,” JHEP 1203, 037 (2012) [arXiv:1110.4382 [hep-th]].
[6] O. Aharony, G. Gur-Ari and R. Yacoby, “Correlation Functions of Large N
Chern-Simons-Matter Theories and Bosonization in Three Dimensions,” JHEP 1212,
028 (2012) [arXiv:1207.4593 [hep-th]].
[7] O. Aharony, “Baryons, monopoles and dualities in Chern-Simons-matter theories,”
JHEP 1602, 093 (2016) [arXiv:1512.00161 [hep-th]].
[8] R. Balian, J. M. Drou↵e and C. Itzykson, “Gauge Fields on a Lattice. 1. General
Outlook,” Phys. Rev. D 10, 3376 (1974); “Gauge Fields on a Lattice. 2. Gauge
Invariant Ising Model,” Phys. Rev. D 11, 2098 (1975); “Gauge Fields on a Lattice. 3.
Strong Coupling Expansions and Transition Points,” Phys. Rev. D 11, 2104 (1975)
Erratum: [Phys. Rev. D 19, 2514 (1979)].
[9] T. Banks and E. Rabinovici, “Finite Temperature Behavior of the Lattice Abelian Higgs
Model,” Nucl. Phys. B 160, 349 (1979).
[10] T. Banks and A. Zaks, “On the Phase Structure of Vector-Like Gauge Theories with
Massless Fermions,” Nucl. Phys. B 196, 189 (1982).
[11] T. Banks and N. Seiberg, “Symmetries and Strings in Field Theory and Gravity,”
Phys. Rev. D 83, 084019 (2011) [arXiv:1011.5120 [hep-th]].
[12] W. A. Bardeen, “Anomalous Ward identities in spinor field theories,” Phys. Rev. 184,
1848 (1969).
[13] M. Barkeshli and J. McGreevy, “Continuous transition between fractional quantum Halland superfluid states,” Phys. Rev. B 89, no. 23, 235116 (2014) [arXiv:1201.4393[cond-mat.str-el]].
[14] A. A. Belavin, A. M. Polyakov, A. S. Schwartz and Y. S. Tyupkin, “Pseudoparticle
Solutions of the Yang-Mills Equations,” Phys. Lett. B 59, 85 (1975)[15] J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics Physique Fizika 1, 195(1964).

– 431 –

[16] J. S. Bell and R. Jackiw, “A PCAC Puzzle: ⇡ ! 2 in the -model”, Nuovo Cim.
A60, 47 (1969)[17] B. A. Bernevig, “Topological Insulators and Topological Superconductors”, Princeton
University Press (2013)[18] A. Bilal, “Lectures on Anomalies,” arXiv:0802.0634 [hep-th].
[19] J. D. Bjorken, “Asymptotic Sum Rules at Infinite Momentum,” Phys. Rev. 179, 1547
(1969).
[20] E. B. Bogomolny, “Stability of Classical Solutions,” Sov. J. Nucl. Phys. 24, 449 (1976)[21] C. Bouchiat, J. Iliopoulos and P. Meyer, “An Anomaly Free Version of Weinberg’s
Model,” Phys. Lett. 38B, 519 (1972).
[22] C. G. Callan, Jr., S. R. Coleman, J. Wess and B. Zumino, “Structure ofphenomenological Lagrangians. 2.,” Phys. Rev. 177, 2247 (1969).
[23] C. G. Callan, Jr., R. F. Dashen and D. J. Gross, “The Structure of the Gauge Theory
Vacuum,” Phys. Lett. B 63, 334 (1976)[24] C. G. Callan, Jr., R. F. Dashen and D. J. Gross, “Instantons and Massless Fermions in
Two-Dimensions,” Phys. Rev. D 16 (1977) 2526.
[25] C. G. Callan, Jr. and J. A. Harvey, “Anomalies and Fermion Zero Modes on Strings
and Domain Walls,” Nucl. Phys. B 250, 427 (1985).
[26] S. R. Coleman, “There are no Goldstone bosons in two-dimensions,” Commun. Math.
Phys. 31, 259 (1973).
[27] S. R. Coleman and E. J. Weinberg, “Radiative Corrections as the Origin of
Spontaneous Symmetry Breaking,” Phys. Rev. D 7, 1888 (1973).
[28] S. R. Coleman, R. Jackiw and L. Susskind, “Charge Shielding and Quark Confinementin the Massive Schwinger Model,” Annals Phys. 93, 267 (1975).
[29] S. R. Coleman, “The Quantum Sine-Gordon Equation as the Massive Thirring Model,”Phys. Rev. D 11, 2088 (1975).
[30] S. R. Coleman, “More About the Massive Schwinger Model,” Annals Phys. 101, 239(1976).
[31] S. R. Coleman and E. Witten, “Chiral Symmetry Breakdown in Large N
Chromodynamics,” Phys. Rev. Lett. 45, 100 (1980).
[32] S. Coleman, “Aspects of Symmetry : Selected Erice Lectures,”, Cambridge University
Press[33] M. Creutz, “Monte Carlo Study of Quantized SU(2) Gauge Theory,” Phys. Rev. D 21,2308 (1980).
[34] M. Creutz, “Quarks, Gluons and Lattices”, Cambridge University Press– 432 –

[35] M. Creutz, “Why rooting fails,” PoS Lattice 2007, 007 (2007) [arXiv:0708.1295[hep-lat]].
[36] A. D’Adda, M. Luscher and P. Di Vecchia, “A 1/N Expandable Series of Nonlinear
Sigma Models with Instantons,” Nucl. Phys. B 146, 63 (1978).
[37] C. Dasgupta and B. I. Halperin, “Phase Transition in a Lattice Model ofSuperconductivity,” Phys. Rev. Lett. 47, 1556 (1981).
[38] R. F. Dashen, “Some features of chiral symmetry breaking,” Phys. Rev. D 3, 1879
(1971).
[39] R. Delbourgo and A. Salam, “The gravitational correction to PCAC,” Phys. Lett. 40B,
381 (1972). doi:10.1016/0370-2693(72)90825-8[40] P. Di Francesco, P. Mathieu and D. Senechal, “Conformal Field Theory,”, Springer[41] P. Di Vecchia and G. Veneziano, “Chiral Dynamics in the Large n Limit,” Nucl. Phys.
B 171, 253 (1980).
[42] S. Deser, R. Jackiw and S. Templeton, “Topologically Massive Gauge Theories,” Annals
Phys. 140, 372 (1982)[43] S. Deser, R. Jackiw and S. Templeton, “Three-Dimensional Massive Gauge Theories,”
Phys. Rev. Lett. 48, 975 (1982).
[44] P. A. M. Dirac, “Quantised Singularities in the Electromagnetic Field”, Proc. R. Soc.
Lond. A, 1931.
[45] P. A. M. Dirac “The Theory of Magnetic Poles”, Physical Review 75, 7 (1948)[46] P. A. M. Dirac, letter to Abdus Salam 1981, published in N. Craigie, P. Goddard and
W. Nahm, “Monopoles in Quantum Field Theory”, Proc. of Monopole Meeting, Trieste,
Italy, 1981.
[47] N. Dombey, “An Interview with Murray Gell-Mann on June 5 2001”, arXiv:2007.12615[48] S. D. Drell, M. Weinstein and S. Yankielowicz, “Strong Coupling Field Theories. 2.
Fermions and Gauge Fields on a Lattice,” Phys. Rev. D 14, 1627 (1976).
[49] G. V. Dunne, “Aspects of Chern-Simons theory,” hep-th/9902115.
[50] H. Eichenherr, “SU(N) Invariant Nonlinear Sigma Models,” Nucl. Phys. B 146, 215(1978) Erratum: [Nucl. Phys. B 155, 544 (1979)].
[51] E. Eichten and J. Preskill, “Chiral Gauge Theories on the Lattice,” Nucl. Phys. B 268,
179 (1986).
[52] S. Elitzur, “Impossibility of Spontaneously Breaking Local Symmetries,” Phys. Rev. D
12, 3978 (1975).
[53] S. Elitzur, G. W. Moore, A. Schwimmer and N. Seiberg, “Remarks on the Canonical
Quantization of the Chern-Simons-Witten Theory,” Nucl. Phys. B 326, 108 (1989).

– 433 –

[54] A. M. Essin, Essin, J. E. Moore and D. Vanderbilt, “Magnetoelectric polarizability andaxion electrodynamics in crystalline insulators”, arXiv:0810.2998 [cond-mat], Phys.
Rev. Lett. 102, 146805 (2009)[55] G. Farmelo, “The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius”,
Faber and Faber, 2009[56] R. P. Feynman, “Very high-energy collisions of hadrons,” Phys. Rev. Lett. 23, 1415
(1969).
[57] R. P. Feynman, “The behavior of hadron collisions at extreme energies,” Conf. Proc. C
690905, 237 (1969).
[58] L. Fidkowski and A. Kitaev, “The e↵ects of interactions on the topological classificationof free fermion systems,” Phys. Rev. B 81, 134509 (2010) [arXiv:0904.2197[cond-mat.str-el]].
[59] L. Fidkowski and A. Kitaev,, “Topological phases of fermions in one dimension”, Phys.
Rev. B 83, 075103 (2011), [arXiv:1008.4138 [cond-mat.str-el]].
[60] D. Finkelstein and J. Rubinstein, “Connection between spin, statistics, and kinks,” J.
Math. Phys. 9, 1762 (1968).
[61] D. H. Lee and M. P. A. Fisher, “Correspondence between two-dimensional bosons and abulk superconductor in a magnetic field” Phys. Rev. B 39, 2756 (1989).
[62] V. Fock, “Über die invariante Form der Wellen- und der Bewegungsgleichungen füreinen geladenen Massenpunkt”, Zeit. für Physik 39, 226-232. (1926)[63] E. H. Fradkin and S. H. Shenker, “Phase Diagrams of Lattice Gauge Theories with
Higgs Fields,” Phys. Rev. D 19, 3682 (1979).
[64] Y. Frishman, A. Schwimmer, T. Banks and S. Yankielowicz, “The Axial Anomaly andthe Bound State Spectrum in Confining Theories,” Nucl. Phys. B 177, 157 (1981).
[65] H. Fritzsch, M. Gell-Mann and H. Leutwyler, “Advantages of the Color Octet Gluon
Picture,” Phys. Lett. 47B, 365 (1973).
[66] L. Fu, C.L. Kane, E.J. Mele, “Topological Insulators in Three Dimensions”, Phys. Rev.
Lett. 98, 106803 (2007)[67] L. Fu and C. L. Kane, “Topological Insulators with Inversion Symmetry”,
cond-mat/0611341, Phys. Rev. B 76, 045302 (2007)[68] K. Fujikawa, “Path Integral Measure for Gauge Invariant Fermion Theories,” Phys.
Rev. Lett. 42, 1195 (1979).
[69] K. Fujikawa, “Path Integral for Gauge Theories with Fermions,” Phys. Rev. D 21, 2848
(1980) Erratum: [Phys. Rev. D 22, 1499 (1980)].
[70] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, “Generalized Global Symmetries,”
JHEP 1502, 172 (2015) [arXiv:1412.5148 [hep-th]].

– 434 –

[71] D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, “Theta, Time Reversal, andTemperature,” JHEP 1705, 091 (2017) [arXiv:1703.00501 [hep-th]].
[72] M. Gell-Mann and M. Levy, “The axial vector current in beta decay,” Nuovo Cim. 16,
705 (1960).
[73] P. H. Ginsparg and K. G. Wilson, “A Remnant of Chiral Symmetry on the Lattice,”
Phys. Rev. D 25, 2649 (1982).
[74] P. H. Ginsparg, “Applied Conformal Field Theory,” hep-th/9108028.
[75] P. H. Ginsparg and G. W. Moore, “Lectures on 2-D gravity and 2-D string theory,”,
hep-th/9304011.
[76] S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S. R. Wadia and X. Yin,
“Chern-Simons Theory with Vector Fermion Matter,” Eur. Phys. J. C 72, 2112 (2012)[arXiv:1110.4386 [hep-th]].
[77] S. L. Glashow, “Partial Symmetries of Weak Interactions,” Nucl. Phys. 22, 579 (1961).
[78] J. Goldstone, “Field Theories with Superconductor Solutions,” Nuovo Cim. 19, 154(1961).
[79] J. Goldstone, A. Salam and S. Weinberg, “Broken Symmetries,” Phys. Rev. 127, 965(1962).
[80] P. Goddard, J. Nuyts and D. I. Olive, “Gauge Theories and Magnetic Charge,” Nucl.
Phys. B 125, 1 (1977).
[81] V. L. Golo and A. M. Perelomov, “Solution of the Duality Equations for the
Two-Dimensional SU(N) Invariant Chiral Model,” Phys. Lett. 79B, 112 (1978).
[82] D. Gross and R. Jackiw, “E↵ect of Anomalies on Quasirenormalisable Theories”, Phys.
Rev. D6, 477 (1972)[83] D. J. Gross and F. Wilczek, “Ultraviolet Behavior of Nonabelian Gauge Theories,”
Phys. Rev. Lett. 30, 1343 (1973).
[84] D. J. Gross and F. Wilczek, “Asymptotically Free Gauge Theories I” Phys. Rev. D 8,3633 (1973).
[85] D. J. Gross and A. Neveu, “Dynamical Symmetry Breaking in Asymptotically Free
Field Theories,” Phys. Rev. D 10, 3235 (1974).
[86] D. J. Gross, “Twenty five years of asymptotic freedom,” Nucl. Phys. Proc. Suppl. 74,
426 (1999) [hep-th/9809060].
[87] F. D. M. Haldane, “Continuum dynamics of the 1-D Heisenberg antiferromagnetic
identification with the O(3) nonlinear sigma model,” Phys. Lett. A 93, 464 (1983).
[88] D. Haldane, “Model for a Quantum Hall E↵ect without Landau Levels: Condensed
Matter Realisation of the Parity Anomaly”, Phys. Rev. Lett. 61, 2015 (1988).

– 435 –

[89] J. A. Harvey, “Magnetic monopoles, duality and supersymmetry,” In *Trieste 1995,
High energy physics and cosmology* 66-125 [hep-th/9603086].
[90] J. A. Harvey, “TASI 2003 lectures on anomalies,” hep-th/0509097.
[91] M. Z. Hasan, C. L. Kane, “Topological Insulators”, arXiv:1002.3895, Rev. Mod. Phys.
82: 3045 (2010)[92] P. Hasenfratz and G. ’t Hooft, “A Fermion-Boson Puzzle in a Gauge Theory,” Phys.
Rev. Lett. 36, 1119 (1976).
[93] G. ’t Hooft, “Renormalization of Massless Yang-Mills Fields,” Nucl. Phys. B 33, 173(1971).
[94] G. ’t Hooft, “Renormalizable Lagrangians for Massive Yang-Mills Fields,” Nucl. Phys.
B 35, 167 (1971).
[95] G. ’t Hooft and M. Veltman “Regularization and Renormalisation of Gauge Fields”,
Nucl. Phys. B44, 189 (1972)[96] G. ’t Hooft, “The Renormalisation Procedure for Yang-Mills Fields”, PhD Thesis
Utrecht University (1972)[97] G. ’t Hooft, “A Planar Diagram Theory for Strong Interactions,” Nucl. Phys. B 72,
461 (1974).
[98] G. ’t Hooft, “A Two-Dimensional Model for Mesons,” Nucl. Phys. B 75, 461 (1974).
doi:10.1016/0550-3213(74)90088-1[99] G. ’t Hooft, “Magnetic Monopoles in Unified Gauge Theories,” Nucl. Phys. B 79, 276(1974).
[100] G. ’t Hooft, “Quarks and Gauge Fields”, Proceedings of the Colloqium on ”Recent
Progress in Lagrangian Field Theory and Applications”, Marseille, June 24-28, 1974
Available here.
[101] G. ’t Hooft, “Symmetry Breaking Through Bell-Jackiw Anomalies,” Phys. Rev. Lett.
37, 8 (1976).
[102] G. ’t Hooft, “Computation of the Quantum E↵ects Due to a Four-Dimensional
Pseudoparticle,” Phys. Rev. D 14, 3432 (1976) Erratum: [Phys. Rev. D 18, 2199
(1978)].
[103] G. ’t Hooft, “On the Phase Transition Towards Permanent Quark Confinement,”
Nucl. Phys. B 138, 1 (1978).
[104] G. ’t Hooft, “A Property of Electric and Magnetic Flux in Non-Abelian Gauge
Theories,” Nucl. Phys. B 153, 141 (1979).
[105] G. ’t Hooft, “Naturalness, chiral symmetry, and spontaneous chiral symmetry
breaking,” NATO Sci. Ser. B 59, 135 (1980).

– 436 –

[106] G. ’t Hooft, “Topology of the Gauge Condition and New Confinement Phases in
Non-Abelian Gauge Theories,” Nucl. Phys. B 190, 455 (1981).
[107] G. ’t Hooft, “When was asymptotic freedom discovered? or the rehabilitation ofquantum field theory,” Nucl. Phys. Proc. Suppl. 74, 413 (1999) [hep-th/9808154].
[108] G. ’t Hooft ed. “50 years of Yang-Mills theory,” Hackensack, USA: World Scientific(2005) 487 p[109] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y.S. Hor, R.J. Cava, M.Z. Hasan, “A topological
Dirac insulator in a quantum spin Hall phase (experimental realization of a 3D
Topological Insulator)”, arXiv:0910.2420 [cond-mat], Nature 452, 970 (2008).
[110] K. A. Intriligator and N. Seiberg, “Mirror symmetry in three-dimensional gauge
theories,” Phys. Lett. B 387, 513 (1996) [hep-th/9607207].
[111] R. Jackiw and C. Rebbi, “Solitons with Fermion Number 1/2,” Phys. Rev. D 13, 3398
(1976).
[112] R. Jackiw and C. Rebbi, “Spin from Isospin in a Gauge Theory,” Phys. Rev. Lett. 36,
1116 (1976).
[113] R. Jackiw and C. Rebbi, “Vacuum Periodicity in a Yang-Mills Quantum Theory,”
Phys. Rev. Lett. 37, 172 (1976).
[114] R. Jackiw and C. Rebbi, “Spinor Analysis of Yang-Mills Theory,” Phys. Rev. D 16,1052 (1977).
[115] R. Jackiw, “Topological Investigations in Quantum Gauge Theories”, in
S. B. Treiman, E. Witten, R. Jackiw and B. Zumino, “Current Algebra And
Anomalies,” Singapore, Singapore: World Scientific ( 1985) 537p
[116] J. D. Jackson and L. B. Okun, “Historical Roots of Gauge Invariance”,
hep-ph/0012061 Rev. Mod. Phys. 73:663-680 (2001)[117] D. B. Kaplan, “A Method for simulating chiral fermions on the lattice,” Phys. Lett. B
288, 342 (1992) [hep-lat/9206013].
[118] D. B. Kaplan, “Chiral Symmetry and Lattice Fermions,” arXiv:0912.2560 [hep-lat].
[119] A. Karch and D. Tong, “Particle-Vortex Duality from 3d Bosonization,” Phys. Rev. X
6, no. 3, 031043 (2016) [arXiv:1606.01893 [hep-th]].
[120] A. Kapustin, “Wilson-’t Hooft operators in four-dimensional gauge theories andS-duality,” Phys. Rev. D 74, 025005 (2006) [hep-th/0501015].
[121] Y. Kikukawa and T. Noguchi, “Low-energy e↵ective action of domain wall fermion
and the Ginsparg-Wilson relation,” Nucl. Phys. Proc. Suppl. 83, 630 (2000)[hep-lat/9902022].
[122] O. Klein, “On the Theory of Charged Fields”, Written for a conference in Conferencein Kazimierz, Poland 1938.

– 437 –

[123] Z. Komargodski, A. Sharon, R. Thorngren and X. Zhou, “Comments on Abelian Higgs
Models and Persistent Order,” arXiv:1705.04786 [hep-th].
[124] J. B. Kogut and L. Susskind, “Hamiltonian Formulation of Wilson’s Lattice Gauge
Theories,” Phys. Rev. D 11, 395 (1975).
[125] J. Kondo, “Resistance Minimum in Dilute Magnetic Alloys,” Prog. Theor. Phys. 32,
no. 1, 37 (1964).
[126] M. Lüscher, “Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation,”
Phys. Lett. B 428, 342 (1998) [hep-lat/9802011].
[127] M. Luscher, “Chiral gauge theories revisited,” Subnucl. Ser. 38, 41 (2002)[hep-th/0102028].
[128] A. Luther and I. Peschel, “Single-particle states, Kohn anomaly, and pairing
fluctuations in one dimension”, Phys. Rev. B9, 7 (1974)[129] J. M. Maldacena, “The Large N limit of superconformal field theories andsupergravity,” Int. J. Theor. Phys. 38, 1113 (1999) [Adv. Theor. Math. Phys. 2, 231(1998)] [hep-th/9711200].
[130] S. Mandelstam, “Vortices and Quark Confinement in Nonabelian Gauge Theories,”
Phys. Rept. 23, 245 (1976).
[131] S. Mandelstam, “Soliton Operators for the Quantized Sine-Gordon Equation,” Phys.
Rev. D 11, 3026 (1975).
[132] A. V. Manohar, “Large N QCD,” hep-ph/9802419.
[133] N. S. Manton and P. Sutcli↵e, “Topological solitons,”, Cambridge University Press(2004)[134] N. D. Mermin and H. Wagner, “Absence of ferromagnetism or antiferromagnetism in
one-dimensional or two-dimensional isotropic Heisenberg models,” Phys. Rev. Lett. 17,1133 (1966).
[135] J. Mcgreevy, “Applied String Theory”,
https://mcgreevy.physics.ucsd.edu/fall08/index.html[136] M. A. Metlitski and A. Vishwanath, “Particle-vortex duality of two-dimensional Diracfermion from electric-magnetic duality of three-dimensional topological insulators,”
Phys. Rev. B 93, no. 24, 245151 (2016) [arXiv:1505.05142 [cond-mat.str-el]].
[137] M. A. Metlitski, “S-duality of u(1) gauge theory with ✓ = ⇡ on non-orientablemanifolds: Applications to topological insulators and superconductors,”
arXiv:1510.05663 [hep-th].
[138] G. Moore, “Informal Lectures on Lattice Gauge Theory”,
https://theorie.ikp.physik.tu-darmstadt.de/qcd/moore/latt lectures.pdf– 438 –

[139] J. E. Moore and L. Balents, “Topological invariants of time-reversal-invariant band
structures”, cond-mat/0607314, Phys. Rev. B 75, 121306(R) (2007)[140] Y. Nambu, “Axial vector current conservation in weak interactions,” Phys. Rev. Lett.
4, 380 (1960).
[141] Y. Nambu and G. Jona-Lasinio, “Dynamical Model of Elementary Particles Based on
an Analogy with Superconductivity. 1.,” Phys. Rev. 122, 345 (1961).
[142] Y. Nambu, “Strings, Monopoles and Gauge Fields,” Phys. Rev. D 10, 4262 (1974).
[143] H. Neuberger, “Exactly massless quarks on the lattice,” Phys. Lett. B 417, 141 (1998)[hep-lat/9707022].
[144] H. Neuberger, “More about exactly massless quarks on the lattice,” Phys. Lett. B 427,
353 (1998) [hep-lat/9801031].
[145] H. B. Nielsen and P. Olesen, “Vortex Line Models for Dual Strings,” Nucl. Phys. B
61, 45 (1973).
[146] H. B. Nielsen and M. Ninomiya, “Absence of Neutrinos on a Lattice. 1. Proof by
Homotopy Theory,” Nucl. Phys. B 185, 20 (1981) Erratum: [Nucl. Phys. B 195, 541(1982)].
[147] H. B. Nielsen and M. Ninomiya, “Absence of Neutrinos on a Lattice. 2. Intuitive
Topological Proof,” Nucl. Phys. B 193, 173 (1981).
[148] N. K. Nielsen, “Asymptotic Freedom as a Spin E↵ect” Am. J. Phys 49;12 1171 (1981)
�[149] A. J. Niemi and G. W. Semeno↵, “Axial
Anomaly Induced Fermion Fractionizationand E↵ective Gauge Theory Actions in Odd Dimensional Space-Times,” Phys. Rev.
Lett. 51, 2077 (1983). doi:10.1103/PhysRevLett.51.2077[150] L. O’Raifeartaigh, “The Dawning of Gauge Theory”, Princeton University Press(1997)[151] W. Pauli, letter to A. Pais entitled “Meson-Nucleon Interaction and Di↵erential
Geometry”, 1953
[152] M. E. Peskin, “Mandelstam ’t Hooft Duality in Abelian Lattice Models,” Annals Phys.
113, 122 (1978).
[153] M. E. Peskin, “Chiral Symmetry And Chiral Symmetry Breaking,” SLAC-PUB-3021.
pdf here.
[154] M. E. Peskin and D. V. Schroeder, “An Introduction to quantum field theory,” CRC
Press[155] H. Poincaré, “Remarques sur une expérience de M. Birkeland, Compt. Rend. Acad.
Sci. Paris 123 (1896), 530-533.

– 439 –

[156] H. D. Politzer, “Reliable Perturbative Results for Strong Interactions?,” Phys. Rev.
Lett. 30, 1346 (1973).
[157] H. D. Politzer, “The dilemma of attribution,” Nobel Lecture, Proc. Nat. Acad. Sci.
102, 7789 (2005) [Int. J. Mod. Phys. A 20, 5741 (2005)][158] A. M. Polyakov, “Particle Spectrum in the Quantum Field Theory,” JETP Lett. 20,
194 (1974) [Pisma Zh. Eksp. Teor. Fiz. 20, 430 (1974)].
[159] A. M. Polyakov, “Compact Gauge Fields and the Infrared Catastrophe,” Phys. Lett. B
59, 82 (1975)[160] A. M. Polyakov, “Quark Confinement and Topology of Gauge Groups,” Nucl. Phys. B
120, 429 (1977).
[161] A. M. Polyakov, “Fermi-Bose Transmutations Induced by Gauge Fields,” Mod. Phys.
Lett. A 3, 325 (1988).
[162] M. K. Prasad and C. M. Sommerfield, “An Exact Classical Solution for the ’t Hooft
Monopole and the Julia-Zee Dyon,” Phys. Rev. Lett. 35, 760 (1975).
[163] J. Preskill and S. Weinberg, “Decoupling Constraints on Massless Composite
Particles,” Phys. Rev. D 24 (1981), 1059
[164] J. Preskill, “Magnetic Monopoles,” Ann. Rev. Nucl. Part. Sci. 34, 461 (1984).
[165] X-L. Qi, T. Hughes and S-C. Zhang, “Topological Field Theory of Time-Reversal
Invariant Insulators”, arXiv:0802.3537 [cond-mat], Phys. Rev. B78 195424 (2008)[166] X-L. Qi, R. Li, J. Zang, S-C. Zhang, “Seeing the magnetic monopole through themirror of topological surface states”, arXiv:0811.1303 [cond-mat] Science 323, 1184
(2009)[167] X-L Qi and S-C Zhang, “Topological insulators and superconductors”, arXiv:1008.2026[cond-mat], Rev. Mod. Phys. 83, 1057-1110 (2011)[168] A. N. Redlich, “Gauge Noninvariance and Parity Violation of Three-Dimensional
Fermions,” Phys. Rev. Lett. 52, 18 (1984).
[169] A. N. Redlich, “Parity Violation and Gauge Noninvariance of the E↵ective Gauge
Field Action in Three-Dimensions,” Phys. Rev. D 29, 2366 (1984).
[170] R. Roy, “Three dimensional topological invariants for time reversal invariant
Hamiltonians and the three dimensional quantum spin Hall e↵ect”, cond-mat/0607531,
Phys. Rev. B 79, 195322 (2009)[171] M.N. Saha, “The Origin of Mass in Neutrons and Protons”, Ind. Jour. Phys., 10, 141,1936
[172] M.N. Saha, “Note on Dirac’s Theory of Magnetic Poles”, Phys. Rev. 951968 (1949)– 440 –

[173] S. Scherer and M. R. Schindler, “A Chiral perturbation theory primer,”
hep-ph/0505265.
[174] J. S. Schwinger, “Gauge Invariance and Mass. 2.,” Phys. Rev. 128, 2425 (1962).
[175] N. Seiberg and E. Witten, “Gapped Boundary Phases of Topological Insulators via
Weak Coupling,” PTEP 2016, no. 12, 12C101 (2016) [arXiv:1602.04251[cond-mat.str-el]].
[176] N. Seiberg, T. Senthil, C. Wang and E. Witten, “A Duality Web in 2+1 Dimensions
and Condensed Matter Physics,” Annals Phys. 374, 395 (2016) [arXiv:1606.01989[hep-th]].
[177] D. Senechal, “An Introduction to bosonization,” cond-mat/9908262.
[178] R. Shankar, “Bosonization: How to make it work for you in condensed matter,” Acta
Phys. Polon. B 26, 1835 (1995).
[179] R. Shankar and G. Murthy, “Deconfinement in d=1: A Closer look,” Phys. Rev. B 72,
224414 (2005) [cond-mat/0508242].
[180] S. R. Sharpe, “Rooted staggered fermions: Good, bad or ugly?,” PoS LAT 2006, 022(2006) [hep-lat/0610094].
[181] R. Shaw, Cambridge PhD Thesis.
[182] M. Shifman, “Instantons in Gauge Theories”, World Scientific (1994)[183] Y. M. Shnir, “Magnetic Monopoles, Springer (2005)[184] T. H. R. Skyrme, “A Nonlinear field theory,” Proc. Roy. Soc. Lond. A 260, 127 (1961).
[185] T. H. R. Skyrme, “A Unified Field Theory of Mesons and Baryons,” Nucl. Phys. 31,
556 (1962).
[186] D. T. Son, “Is the Composite Fermion a Dirac Particle?,” Phys. Rev. X 5, no. 3,031027 (2015) [arXiv:1502.03446 [cond-mat.mes-hall]].
[187] J. Steinberger, “On the Use of subtraction fields and the lifetimes of some types ofmeson decay,” Phys. Rev. 76, 1180 (1949).
[188] L. Susskind, “Lattice Fermions,” Phys. Rev. D 16, 3031 (1977).
[189] S. Raby, S. Dimopoulos and L. Susskind, “Tumbling Gauge Theories,” Nucl. Phys. B
169, 373 (1980).
[190] D. G. Sutherland, “Current algebra and some nonstrong mesonic decays,” Nucl. Phys.
B 2, 433 (1967).
[191] D. Tong, “TASI Lectures on Instantons”, hep-th/0509216
http://www.damtp.cam.ac.uk/user/tong/tasi.html[192] D. Tong, “Lectures on String Theory”, arXiv:0908.0333
http://www.damtp.cam.ac.uk/user/tong/string.html– 441 –

[193] D. Tong, “Lectures on the Quantum Hall E↵ect”, arXiv:1606.06687
http://www.damtp.cam.ac.uk/user/tong/qhe.html[194] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall
Conductance in a Two-Dimensional Periodic Potential”, Phys. Rev. Lett. 49, 405(1982).
[195] C. Vafa and E. Witten, “Restrictions on Symmetry Breaking in Vector-Like Gauge
Theories,” Nucl. Phys. B 234, 173 (1984).
[196] C. Vafa and E. Witten, “Parity Conservation in QCD,” Phys. Rev. Lett. 53, 535(1984).
[197] S, Vandoren and P. van Nieuwenhuizen, “Lectures on Instantons”, arXiv:0802.1862[hep-th][198] M. Veltman “Theoretical aspects of high energy neutrino interactions”, Proc. Roy.
Soc. A 301, 1465
[199] G. Veneziano, “U(1) Without Instantons,” Nucl. Phys. B 159, 213 (1979).
[200] C. Wang and T. Senthil, “Half-filled Landau level, topological insulator surfaces, andthree-dimensional quantum spin liquids,” Phys. Rev. B 93, no. 8, 085110 (2016)[arXiv:1507.08290 [cond-mat.str-el]].
[201] F. J. Wegner, “Duality in Generalized Ising Models and Phase Transitions Without
Local Order Parameters,” J. Math. Phys. 12, 2259 (1971).
[202] X. G. Wen, “A lattice non-perturbative definition of an SO(10) chiral gauge theory
and its induced standard model,” Chin. Phys. Lett. 30, 111101 (2013) [arXiv:1305.1045[hep-lat]].
[203] H. Weyl, “Gruppentheorie und Quantenmechanik, Leipzig, Hirzel (1928)[204] H. Weyl, “”Gravitation and the electron”, Proc. Nat. Acad. Sci. 15, 323 (1929)[205] S. Weinberg, “A Model of Leptons,” Phys. Rev. Lett. 19, 1264 (1967).
[206] S. Weinberg, “Nonlinear realizations of chiral symmetry,” Phys. Rev. 166, 1568
(1968). doi:10.1103/PhysRev.166.1568[207] S. Weinberg, “Nonabelian Gauge Theories of the Strong Interactions,” Phys. Rev.
Lett. 31, 494 (1973).
[208] S. Weinberg, “Current algebra and gauge theories. 2. NonAbelian gluons,” Phys. Rev.
D 8, 4482 (1973).
[209] S. Weinberg, “E↵ective Field Theory, Past and Future,” PoS CD 09, 001 (2009)[arXiv:0908.1964 [hep-th]].
[210] D. Weingarten, “Mass Inequalities for QCD,” Phys. Rev. Lett. 51 (1983), 1830

– 442 –

[211] F. Wilczek and A. Zee, “Linking Numbers, Spin, and Statistics of Solitons,” Phys.
Rev. Lett. 51, 2250 (1983).
[212] F. Wilczek, “Two Applications of Axion Electrodynamics,” Phys. Rev. Lett. 58, 1799
(1987).
[213] H. Wilson, “Note on Dirac’s Theory of Magnetic Poles”, Phys. Rev. 75, 309 (1949)[214] K. G. Wilson, “Confinement of Quarks,” Phys. Rev. D 10, 2445 (1974).
[215] K. G. Wilson, “The Renormalization Group: Critical Phenomena and the Kondo
Problem,” Rev. Mod. Phys. 47, 773 (1975).
[216] E. Witten, “Dyons of Charge e✓/2⇡,” Phys. Lett. B 86, 283 (1979)[217] E. Witten and D. I. Olive, “Supersymmetry Algebras That Include Topological
Charges,” Phys. Lett. 78B, 97 (1978).
[218] E. Witten, “Some Properties of the ( ¯ )2 Model in Two-Dimensions,” Nucl. Phys. B
142, 285 (1978).
[219] E. Witten, “Chiral Symmetry, the 1/N Expansion, and the SU(N) Thirring Model,”Nucl. Phys. B 145, 110 (1978).
[220] E. Witten, “Instantons, the Quark Model, and the 1/n Expansion,” Nucl. Phys. B
149, 285 (1979).
[221] E. Witten, “Baryons in the 1/N Expansion,” Nucl. Phys. B 160, 57 (1979).
[222] E. Witten, “Current Algebra Theorems for the U(1) Goldstone Boson,” Nucl. Phys. B
156, 269 (1979).
[223] E. Witten, “The 1/N Expansion In Atomic And Particle Physics,” NATO Sci. Ser. B
59, 403 (1980).
[224] E. Witten, ”Large N Chiral Dynamics,” Annals Phys. 128, 363 (1980).
[225] E. Witten, “An SU(2) Anomaly,” Phys. Lett. B 117, 324 (1982)[226] E. Witten, “Current Algebra, Baryons, and Quark Confinement,” Nucl. Phys. B 223,
433 (1983).
[227] E. Witten, “Nonabelian Bosonization in Two-Dimensions,” Commun. Math. Phys.
92, 455 (1984).
[228] E. Witten, “Quantum Field Theory and the Jones Polynomial,” Commun. Math.
Phys. 121, 351 (1989).
[229] E. Witten, “Three Lectures on Topological Phases” arXiv:1510.07698, La Rivista delNuovo Cimento, 39 (2016) 313-370[230] E. Witten, “Fermion Path Integrals And Topological Phases”, arXiv:1508.04715, Rev.
Mod. Phys. 88, 35001 (2016)– 443 –

[231] T. T. Wu and C. N. Yang, “Concept of Nonintegrable Phase Factors and Global
Formulation of Gauge Fields,” Phys. Rev. D 12, 3845 (1975).
[232] C. N. Yang and R. L. Mills, “Conservation of Isotopic Spin and Isotopic Gauge
Invariance”, Phys. Rev. 96 (1954) 191-195[233] K. Yee, “Numerical solution of initial boundary value problems involving maxwell’sequations in isotropic media”, IEEE Transactions on Antennas and Propagation, Vol
14, 3, May 1966.
[234] Y. You, Y. BenTov and C. Xu, “Interacting Topological Superconductors and possible
Origin of 16n Chiral Fermions in the Standard Model,” arXiv:1402.4151[cond-mat.str-el].
[235] D. Zwanziger, “Quantum field theory of particles with both electric and magnetic
charges,” Phys. Rev. 176, 1489 (1968).

– 444 –