# Canonical quantization of lattice Chern-Simons theory

Theodore<EMAIL_ADDRESS>and Tin
<EMAIL_ADDRESS>

1Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and
Astronomy,
University of California, Los Angeles, CA 90095, USA
2Department of Mathematical Sciences, Durham University, DH1 3LE Durham, UK

###### Abstract

We discuss the canonical quantization of $U(1)_{k}$ Chern-Simons theory on a
spatial lattice. In addition to the usual local Gauss law constraints, the
physical Hilbert space is defined by 1-form gauge constraints implementing the
compactness of the $U(1)$ gauge group, and (depending on the details of the
spatial lattice) non-local constraints which project out unframed Wilson
loops. Though the ingredients of the lattice model are bosonic, the physical
Hilbert space is finite-dimensional, with exactly $k$ ground states on a
spatial torus. We quantize both the bosonic (even level) and fermionic (odd
level) theories, describing in detail how the latter depends on a choice of
spin structure.

###### Contents

  1. 1 Introduction
  2. 2 Warmup: the non-compact theory
    1. 2.1 Symmetries and gauge redundancies
  3. 3 Compact Chern-Simons theory
    1. 3.1 Quantization
    2. 3.2 Gauge redundancies
    3. 3.3 $\mathbb{Z}_{k}$ 1-form symmetry and ’t Hooft anomaly
    4. 3.4 Defect Hilbert spaces and topological spin
  4. 4 Odd levels: the fermionic Chern-Simons theory
  5. 5 Conclusions
  6. A Quantizing a first-order Lagrangian

##  1 Introduction

Chern-Simons (CS) theory is the quintessential topological quantum field
theory (TQFT), with numerous applications and appearances throughout condensed
matter and high-energy physics. Even the simplest abelian CS theory plays a
prevalent role in the study of anomalies, level/rank and boson/fermion
dualities, defects in higher-dimensional theories, and in the quantum field
theories of spin liquids and the fractional quantum hall effect.

The literature is host to a multitude [1, 2, 3, 4, 5, 6, 7, 8] of distinct
formulations of Chern-Simons theory on the lattice, whose variety reflects the
lack of universality familiar from continuum quantum field theory. In a
previous paper [9] we realized $U(1)_{k}$ CS theory as a lattice gauge theory
on a Euclidean spacetime lattice using the modified Villain formalism [10,
11]. This framework can be used to endow lattice models with key features at
finite lattice spacing which in a conventional discretization would only
emerge in the continuum limit.333See e.g. Refs. [12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24] for applications of this formalism, and especially
Refs. [22, 23, 24] which employ the modified Villain approach to ‘doubled’ CS
theory on the lattice.  The present paper follows up on our previous work [9]
and is dedicated to the canonical quantization of abelian CS theory on a
spatial lattice within the Villain Hamiltonian [17] approach. Given the long
list of works on this subject, one may wonder what new insights can be gained
by revisiting this problem. We find that the lattice Hamiltonian formulation
provides useful perspectives on certain crucial features of CS theory which
were preserved by our spacetime discretization, such as compactness of the
gauge group, level quantization, the framing of Wilson lines, the anomalous
1-form symmetry, and more.

One of the main advantages of working in the Hamiltonian formulation is that
we have direct access to the Hilbert space. The building blocks of our lattice
CS theory are gauge fields on links and discrete integer fields on plaquettes,
which are each associated with infinite-dimensional Hilbert spaces. However,
it is well-known that the Hilbert space of CS theory in the continuum has a
finite dimension equal to $k^{g}$, where $k$ is the level and $g$ is the genus
of the spatial manifold. This apparent discrepancy is resolved in the
following way: the physical Hilbert space of our lattice model is actually a
projection of the infinite-dimensional Hilbert space down to a finite-
dimensional constrained Hilbert space. There are three types of constraints:
the local Gauss law implementing ‘small’ gauge transformations, a discrete
1-form Gauss law implementing ‘large’ gauge transformations, and a non-local
constraint which projects out unframed Wilson lines. Together, these
constraints imply that the only non-trivial operators are framed, topological
Wilson loops, whose algebra results in a physical Hilbert space whose
dimension matches that of the continuum theory exactly.

In addition to repeating much of the analysis of Ref. [9] in the canonical
formalism, we go further and discuss how to formulate the odd-level CS
theories, which in the continuum are spin-TQFTs requiring a choice of (and
exhibiting a dependence on) spin structure. Using the method of ‘fermion
condensation’ well-known in the TQFT literature [25, 26], we construct a
consistent Hilbert space for the odd-$k$ CS theories and show that Wilson
loops have the spin structure-dependence expected from the continuum. In our
view this comprises a significant step towards establishing nonperturbative
lattice-level fermion-boson dualities, and of placing the decades-old idea of
flux-attachment [27, 28] on equal footing with particle-vortex duality, which
originated on the lattice [29, 30]. For similar work in this direction but
without explicit Chern-Simons terms on the lattice, see Refs. [31, 32].

Figure 1: The physical observables in the compact $U(1)$ Chern-Simons theory
on the lattice are framed Wilson lines, or ribbon operators. The ordinary
Wilson lines at the two edges of the ribbon are connected by a surface
operator built from integer-spectrum Villain fields, such that the ribbons as
a whole are gauge-invariant and topological — they only depend on the homology
class of the line $\tilde{C}$ on the dual lattice which defines their support.

Much of the literature on lattice CS theory was concerned by the presence of
additional gauge field zero modes in the most obvious discretization of the
continuum CS term. As summarized in Ref. [9], these zero modes turn out to be
crucial, rather than detrimental, as they capture the need for framing — i.e.
the fact that the physical Wilson lines in CS theory are ribbons rather than
lines [33]. As mentioned above and discussed in more detail below, these zero
modes are associated with additional constraints which must be imposed on the
Hilbert space which project out all unframed Wilson lines. The need for
additional constraints was noticed in Refs. [3, 4] but deemed unphysical —
these authors gave an alternative discretization of (continuum time) abelian
CS theory without zero modes (see Ref. [7] for the generalization to an
arbitrary graph). Roughly speaking these constructions correspond to choosing
a framing in the time direction. In contrast, we take the view that the zero
modes are a crucial part of the quantization procedure, and our resulting
lattice theory makes manifest certain global aspects which are obscured in
these alternative formulations.

We begin in Sec. 2 by discussing the non-compact CS theory with gauge group
$\mathbb{R}$. The zero modes and non-local constraints leading to framing are
already present in this simple starting point. In Sec. 3 we introduce discrete
Villain fields to gauge a subgroup of the 1-form symmetry of the non-compact
model, making the gauge group $U(1)=\mathbb{R}/2\pi\mathbb{Z}$. Without
additional ingredients, consistency of the gauging procedure quantizes the
level $k$ to be an even integer. Canonically conjugate to the Villain field is
a compact scalar which plays the role of the monopole operator, and carries
electric charge $k$. We discuss the algebra of Wilson loops, its relation to
the ’t Hooft anomaly of the $\mathbb{Z}_{k}$ 1-form symmetry, and the
implications for ground state degeneracy. Then in Sec. 4 we introduce a pair
of Majorana fermions on each plaquette, and use them to consistently quantize
the odd level theories in a way which depends on the spin structure of the
spatial lattice. We conclude in Sec. 5 with directions for future work.

##  2 Warmup: the non-compact theory

Consider Hamiltonian lattice gauge theory [34] formulated on a rectangular
spatial lattice $\Lambda$ where time is real, continuous, and non-compact. We
work on a spatial lattice with $L_{1}$ and $L_{2}$ sites in each direction and
periodic boundary conditions, so that the spatial topology is that of a torus.
We denote the sites, links, and plaquettes by $x,\ell,p$, sometimes using
vectorial notation so that $x=\mathbf{x}$ is the site at $\mathbf{x}$,
$\ell=(\mathbf{x},i)$ is the link starting at $\mathbf{x}$ pointing in the $i$
direction, and $p=(\mathbf{x},12)$ is the plaquette whose lower left corner is
at $\mathbf{x}$. Throughout the paper we use the language of cochains and
lattice differential forms. Functions on sites, links, and plaquettes are
called 0-, 1-, and 2-cochains, with $C^{i}(R)$ denoting the set of $R$-valued
$i$-cochains. We make heavy use of cup products, which are the lattice analogs
of wedge products in de Rham cohomology. We stick to the conventions of Refs.
[11, 9] — see those references for further details.444We will repeatedly use
the following crucial formulas — one is the analog of the Leibniz rule, or
‘summation by parts’
$d(\alpha\cup\beta)=d\alpha\cup\beta+(-1)^{p}\,\alpha\cup d\beta,$ (1) and the
other is an identity capturing the lack of super-commutativity of cup
products:
$\alpha\cup\beta-(-1)^{pq}\,\beta\cup\alpha=(-1)^{p+q+1}\left[d(\alpha\cup_{1}\beta)-d\alpha\cup_{1}\beta-(-1)^{p}\,\alpha\cup_{1}\beta\right]\,,$
(2) where $\alpha,\beta$ are $p-$ and $q-$cochains respectively. Here
$\cup_{1}$ is the cup-1 product — the $\cup_{i}$ products vanish unless $i\leq
p,q$. For general formulae for and applications of (higher) cup products on
the (hyper) cubic lattice, see Ref. [35].

Our starting point is a real link field (i.e. a real-valued 1-cochain)
$a_{\ell}$ together with a real field $(a_{0})_{x}$ on sites (i.e. a real-
valued 0-cochain) representing the spatial and time components of the non-
compact $\mathbb{R}$ gauge field. We begin with the lattice action

$S=\frac{k}{4\pi}\int
dt\sum_{p\in\Lambda}\Big{[}(a\cup(\dot{a}-da_{0}))_{p}-(a_{0}\cup
da)_{p}\Big{]}\;,$ (3)

with $k$ an arbitrary real number. Summing by parts, the above can also be
written as

$S=\frac{k}{4\pi}\int dt\sum_{p\in\Lambda}\Big{[}(a\cup\dot{a})_{p}-(da\cup
a_{0}+a_{0}\cup da)_{p}\Big{]}\;.$ (4)

In the standard lattice notation it is

$S=\frac{k}{4\pi}\int
dt\sum_{\mathbf{x}}\Big{[}\epsilon^{ij}a_{\mathbf{x},i}\,\dot{a}_{\mathbf{x}+\hat{i},j}-(a_{0})_{\mathbf{x}}\left((da)_{\mathbf{x}-\mathbf{s},12}+(da)_{\mathbf{x},12}\right)\Big{]}\;,$
(5)

where we have defined $\mathbf{s}\equiv\hat{1}+\hat{2}$ and the sum over
$i,j=1,2$ is implicit. Now consider the gauge transformation

$a\rightarrow a+d\lambda\,,\ a_{0}\rightarrow a_{0}+\dot{\lambda}\,.$ (6)

where $\lambda$ is a real valued 0-cochain with arbitrary time dependence. The
action is invariant under the above gauge transformation (which is easiest to
see in the cup-product formulation). Explicitly,

$\Delta S=\frac{k}{4\pi}\int dt\sum_{\Lambda}\left[d\lambda\cup\dot{a}+a\cup
d\dot{\lambda}+d\lambda\cup
d\dot{\lambda}-da\cup\dot{\lambda}-\dot{\lambda}\cup da\right]\\\
=\frac{k}{4\pi}\int dt\sum_{\Lambda}\left[\partial_{t}(d\lambda\cup
a)-d(\dot{\lambda}\cup a)-d(a\cup\dot{\lambda})+d(\lambda\cup
d\dot{\lambda})\right]\,,$ (7)

using the Leibniz rule. Since we take time to be non-compact and the spatial
lattice has no boundary, the gauge variation vanishes.

From the above action we see that the canonical momentum of $a_{0}$ is zero,
forcing a Gauss constraint555The Hamiltonian would be proportional to
$a_{0}(\dots)$, and since the canonical momentum of $a_{0}$ has to commute
with the Hamiltonian, we must have that $(\dots)=0$.

$\mathcal{G}_{\mathbf{x}}=(da)_{\mathbf{x}-\mathbf{s},12}+(da)_{\mathbf{x},12}=0\,.$
(8)

Note that this is not what we expect in the CS theory in the continuum, where
the flux of $a$ is set to zero, which renders Wilson lines topological. The
reason for the above constraint instead of $da=0$ is the lack of graded
commutativity of the cup product.

We proceed with the quantization. We write the action in terms of the momentum
space fields

$a_{\mathbf{x},i}=\frac{1}{\sqrt{V}}\sum_{\mathbf{p}}\tilde{a}_{\mathbf{p},i}\,e^{i\mathbf{p}\cdot\mathbf{x}}\;,$
(9)

where $\tilde{a}_{\mathbf{p},i}=\tilde{a}_{-\mathbf{p},i}^{\dagger}$, and
$V=L_{1}L_{2}$ is the spatial volume. The action becomes

$S=\frac{k}{4\pi}\int
dt\sum_{\mathbf{p}}\epsilon_{ij}\,\tilde{a}_{\mathbf{p},i}\dot{\tilde{a}}_{-\mathbf{p},j}\,e^{-i\mathbf{p}_{i}}+\cdots,$
(10)

where the dots indicate terms that vanish when the Gauss constraint is
imposed. We can integrate by parts to write this as

$S=\frac{k}{8\pi}\int
dt\sum_{\mathbf{p}}\epsilon_{ij}(e^{-ip_{i}}+e^{ip_{j}})\,\tilde{a}_{\mathbf{p},i}\dot{\tilde{a}}_{-\mathbf{p},j}=\frac{1}{2}\int
dt\sum_{\mathbf{p}}K(\mathbf{p})_{ij}\,\tilde{a}_{\mathbf{p},i}\dot{\tilde{a}}_{-\mathbf{p},j}\,,$
(11)

where

$K(\mathbf{p})_{ij}=\frac{k}{4\pi}\begin{pmatrix}0&e^{-ip_{1}}+e^{ip_{2}}\\\
-e^{ip_{1}}-e^{-ip_{2}}&0\end{pmatrix}=\frac{k}{4\pi}\epsilon_{ij}(e^{-ip_{i}}+e^{ip_{j}})$
(12)

obeys the antisymmetry property $K(\mathbf{p})_{ij}=-K(-\mathbf{p})_{ji}$.
Since the action is first order in time derivatives, we can simply read off
that the conjugate momentum to $\tilde{a}_{\mathbf{p},i}$ is
$K(-\mathbf{p})_{ji}\tilde{a}_{-\mathbf{p},j}=-K(\mathbf{p})_{ij}\tilde{a}_{-\mathbf{p},j}$.666This
conclusion also follows from a more careful analysis using Dirac brackets. In
this approach one has second-class constraints relating momenta and
coordinates
$\Pi_{\mathbf{p},i}=-\tfrac{1}{2}K(\mathbf{p})_{ij}\tilde{a}_{\mathbf{p},j}$,
along with the Gauss law constraint. See Appendix A for more details, along
with an example of this approach in a simpler setting. The commutation
relations would be obtained by inverting $K(\mathbf{p})_{ij}$. However,
$K(\mathbf{p})_{ij}$ vanishes on a special set of momenta, namely $\mathbf{p}$
such that $p_{1}+p_{2}=\pi$ mod $2\pi$, making it singular. As we will
repeatedly refer to these special momentum modes throughout the paper, it will
be convenient to define

$\mathcal{P}_{\text{staggered}}=\\{\mathbf{p}\,|\,p_{1}+p_{2}=\pi\text{ mod
}2\pi\\}\,.$ (13)

The size of this set depends sensitively on the lengths of the torus, and is
equal to

$|\mathcal{P}_{\text{staggered}}|=\begin{cases}\gcd(L_{1},L_{2})&\,\text{ if
$\text{lcm}(L_{1},L_{2})$ is even,}\\\ 0&\,\text{ if $\text{lcm}(L_{1},L_{2})$
is odd.}\end{cases}$ (14)

If $\mathbf{p}\in\mathcal{P}_{\text{staggered}}$, the vanishing of
$K(\mathbf{p})_{ij}$ simply means that the conjugate momentum for
$\tilde{a}_{\mathbf{p},i}$ vanishes,

$\Pi_{\mathbf{p},i}=0,\ \mathbf{p}\in\mathcal{P}_{\text{staggered}}\,.$ (15)

In the following we will refer to this constraint as the ‘framing constraint,’
for reasons which will become clear soon.

We can proceed with the modes for which
$\mathbf{p}\not\in\mathcal{P}_{\text{staggered}}$. For these modes,
$K(\mathbf{p})_{ij}$ is invertible,

$(K(\mathbf{p})^{-1})_{ij}=-\frac{1}{2}\left(\frac{4\pi}{k}\right)^{2}\frac{1}{1+\cos(p_{1}+p_{2})}K(\mathbf{p})_{ij}\,,$
(16)

so the commutation relations are given by

$[\tilde{a}_{\mathbf{p},i},\tilde{a}_{\mathbf{q},j}]=i\delta_{\mathbf{p},-\mathbf{q}}(K(-\mathbf{p})^{-1})_{ij}=-\frac{2\pi
i}{k}\epsilon_{ij}\,\delta_{\mathbf{p},-\mathbf{q}}\,\frac{e^{ip_{i}}+e^{-ip_{j}}}{1+\cos(p_{1}+p_{2})}\,.$
(17)

In the quantum theory, the physical Hilbert space is defined via the remaining
constraints: the Gauss law (8) and the framing constraint (15). The Gauss law
plays the familiar role as the generator of gauge transformations, which act
trivially on the physical Hilbert space. As a result, gauge non-invariant
operators are projected out from the theory. In momentum space, the Gauss law
reads

$\tilde{\mathcal{G}}_{\mathbf{p}}=(1+e^{i(p_{1}+p_{2})})\epsilon_{ij}(1-e^{ip_{j}})\tilde{a}_{\mathbf{p},i}=0\,,$
(18)

which clearly commutes with the constraints $\Pi_{\mathbf{p},i}=0$ for
$\mathbf{p}\in\mathcal{P}_{\text{staggered}}$ because precisely these modes
are not constrained by the Gauss law. Hence these constraints are first-class
and we can consistently set the canonical momenta for the
$\mathcal{P}_{\text{staggered}}$ modes to zero.

When the canonical momentum $\Pi$ of some coordinate $X$ vanishes, only states
which are translationally invariant in $X$ contribute. In other words, in such
a situation the physical states can all be thought of as $\int
dX\,\left|X\right\rangle$ where $\left|X\right\rangle$ are eigenstates of the
operator $X$. This means that the expectation value of operators of the type
$e^{i\alpha X}$ with $\alpha\in\mathbb{R}$ will vanish.

A generic local operator $O_{\mathbf{x}}$ will contain the peculiar modes with
$\mathbf{p}\in\mathcal{P}_{\text{staggered}}$. However it is easy to project
out these modes by instead considering the sum
$O_{\mathbf{x}}+O_{\mathbf{x}+\mathbf{s}}$. The Fourier modes of this operator
are

$\tilde{O}_{\mathbf{p}}(1+e^{ip_{1}+ip_{2}})e^{i\mathbf{p}\cdot\mathbf{x}}$
(19)

which identically vanishes whenever
$\mathbf{p}\in\mathcal{P}_{\text{staggered}}$. Moreover, the expectation value
of any operator $e^{iO_{\mathbf{x}}}$ will generically vanish because of the
peculiar modes, but $e^{iO_{\mathbf{x}}+iO_{\mathbf{x}+\mathbf{s}}}$ will not.

Figure 2: A framed Wilson line in the non-compact $\mathbb{R}$ Chern-Simons
theory. Here we show the curve $\tilde{C}$ on the dual lattice and its
Poincaré dual $\star[\tilde{C}]$.

To see the connection to framing, let us apply this logic to Wilson loops. A
naive Wilson loop $W(C)$ on a generic contour $C$ contains the
$\mathcal{P}_{\text{staggered}}$ modes and is a vanishing operator on the
Hilbert space! However, the _framed_ Wilson loop

$e^{i\frac{q}{2}\sum_{(\mathbf{x},i)\in
C}(a_{\mathbf{x},i}+a_{\mathbf{x}+\mathbf{s},i})}\,,$ (20)

does not contain the $\mathcal{P}_{\text{staggered}}$ modes and is a genuine
operator in the theory. The framing constraint is so named because it projects
out all unframed Wilson loops, leaving framed loops as the only physical
operators (which, as we will see soon, are also topological). We can write a
general framed Wilson loop in a simple way using cup products. Let $\tilde{C}$
be a contour on the dual lattice, with $(\star[\tilde{C}])_{\ell}$ its
Poincaré dual 1-cochain which is equal to the number of oriented times
$\tilde{C}$ crosses the link $\ell$. The framed Wilson loop is simply

$\widehat{W}_{q}(\tilde{C})=e^{i\frac{q}{2}\sum\star[\tilde{C}]\cup
a-a\cup\star[\tilde{C}]}\,,$ (21)

where the sum is over all plaquettes of the lattice. Since we are in the non-
compact $\mathbb{R}$ gauge theory, $q$ can be an arbitrary real charge. See
Fig. 2 for an example of a framed Wilson line.

The number of vanishing canonical momenta depends sensitively on the lengths
of the torus $L_{1},L_{2}$, and one may wonder whether the requirement that
Wilson loops be framed also depends on the torus geometry. Such dependence
would be rather unphysical. The more precise statement is that the framing
constraint sets to zero any unframed Wilson loop _which cannot be written in
terms of framed Wilson loops_. For instance, if $L_{1}$ and $L_{2}$ are odd,
$\mathcal{P}_{\text{staggered}}$ is empty, and there are no vanishing
canonical momenta — correspondingly, there are ordinary Wilson loops in this
theory. The crucial point is that in this case, each ordinary unframed Wilson
line can be written as a product of framed Wilson lines. We show an example in
Fig. 3. The upshot is that regardless of the geometry of the torus, the
physical operators in the theory are framed Wilson loops, with the framing
dictated by our convention for the cup products in Eq. (3).

Figure 3: On a torus with odd side lengths $L_{1},L_{2}$, any ‘unframed’
Wilson line can be written as a product of $\text{lcm}(L_{1},L_{2})$ framed
Wilson lines displaced by $\mathbf{s}=\hat{1}+\hat{2}$. In the above example,
$L_{1}=L_{2}=5$.

Let us now return to position space and write down the commutation relations
between link fields. The commutator between two links is in fact non-local.
But the commutator of a single link with a _pair_ of links shifted by
$\mathbf{s}$ is ultra-local and given by

$\begin{split}[a_{\mathbf{x},i},a_{\mathbf{y},j}+a_{\mathbf{y}+\mathbf{s},j}]&=\frac{1}{V}\sum_{\mathbf{p},\mathbf{q}}[\tilde{a}_{\mathbf{p},i},\tilde{a}_{\mathbf{q},j}](1+e^{i(q_{1}+q_{2})})e^{i\mathbf{p}\cdot\mathbf{x}+i\mathbf{q}\cdot\mathbf{y}}\\\
&=-\frac{2\pi
i}{k}\frac{1}{V}\sum_{\mathbf{p}}\epsilon_{ij}\frac{e^{ip_{i}}+e^{-ip_{j}}}{1+\cos(p_{1}+p_{2})}(1+e^{-i(p_{1}+p_{2})})e^{i\mathbf{p}\cdot(\mathbf{x}-\mathbf{y})}\\\
&=-\frac{4\pi
i}{k}\frac{1}{V}\sum_{\mathbf{p}}\epsilon_{ij}\,e^{-ip_{j}}e^{i\mathbf{p}\cdot(\mathbf{x}-\mathbf{y})}=-\frac{4\pi
i}{k}\epsilon_{ij}\,\delta_{\mathbf{x},\mathbf{y}+\hat{j}}\,.\end{split}$ (22)

As a result, the commutator between two ‘point-split’ pairs of links is given
by

$[\frac{a_{\mathbf{x},i}+a_{\mathbf{x}+\mathbf{s},i}}{2},\frac{a_{\mathbf{y},j}+a_{\mathbf{y}+\mathbf{s},j}}{2}]=-\frac{2\pi
i}{k}\epsilon_{ij}\frac{\delta_{\mathbf{x}+\hat{i},\mathbf{y}}+\delta_{\mathbf{x},\mathbf{y}+\hat{j}}}{2}\,,$
(23)

which is the lattice analogue of the continuum commutation relation

$[a_{i}(\mathbf{x}),a_{j}(\mathbf{y})]=-\frac{2\pi
i}{k}\epsilon_{ij}\,\delta(\mathbf{x}-\mathbf{y})\,.$ (24)

This leads to the following crucial formula involving an arbitrary pair of
1-cochains $X$ and $Y$:

$\left[\frac{i}{2}\sum_{\Lambda}(X\cup a-a\cup
X)\,,\;\frac{i}{2}\sum_{\Lambda}(Y\cup a-a\cup Y)\right]=\frac{2\pi
i}{2k}\sum_{\Lambda}(X\cup Y-Y\cup X)\,,$ (25)

where the sum is over all plaquettes on the lattice. If we let
$X=\star[\tilde{C}_{1}]$ and $Y=\star[\tilde{C}_{2}]$, the above commutation
relation implies

$\left[\frac{i}{2}\sum_{\Lambda}(\star[\tilde{C}_{1}]\cup
a-a\cup\star[\tilde{C}_{1}])\,,\;\frac{i}{2}\sum_{\Lambda}(\star[\tilde{C}_{2}]\cup
a-a\cup\star[\tilde{C}_{2}])\right]\\\ =\frac{2\pi
i}{2k}\sum_{\Lambda}(\star[\tilde{C}_{1}]\cup\star[\tilde{C}_{2}]-\star[\tilde{C}_{2}]\cup\star[\tilde{C}_{1}])=\frac{2\pi
i}{k}\text{Int}(\tilde{C}_{1},\tilde{C}_{2}).$ (26)

As a result, framed Wilson loops obey the expected algebra

$\widehat{W}_{q_{1}}(\tilde{C}_{1})\widehat{W}_{q_{2}}(\tilde{C}_{2})=\widehat{W}_{q_{2}}(\tilde{C}_{2})\widehat{W}_{q_{1}}(\tilde{C}_{1})=e^{-\frac{2\pi
i}{k}q_{1}q_{2}\text{Int}(\tilde{C}_{1},\tilde{C}_{2})}\,.$ (27)

Figure 4:  Commutation relations (22) between a single link field and a pair
of links displaced by $\mathbf{s}=\hat{1}+\hat{2}$.

###  2.1 Symmetries and gauge redundancies

The action (3) is invariant under shifts of the gauge field of the form
$a\rightarrow a+\omega$ and $a_{0}\rightarrow a_{0}+\omega_{0}$ with the
1-cochain $\omega\in C^{1}(\mathbb{R})$ and the 0-cochain $\omega_{0}\in
C^{0}(\mathbb{R})$ satisfying

$d\omega=0\,,\ d\omega_{0}-\dot{\omega}=0\,.$ (28)

Now let us focus for the moment on the global transformation for which
$\omega_{0}=0$ and $\omega$ is time-independent. Then we can associate
$\omega\in Z^{1}(\mathbb{R})$ and hence we can think of the Poincaré dual of
$\omega$ as a (or some collection of) closed contour(s) $\tilde{C}$ on the
dual lattice. Further we will assume that $\omega$ does not have Fourier modes
in $\mathcal{P}_{\text{staggered}}$, because shifting these modes is
automatically a symmetry since all the states must have a vanishing canonical
momentum of these modes.

The generator of this transformation is simply

$U[\omega]=e^{\frac{ik}{4\pi}\sum_{\mathbf{x}}\omega_{\mathbf{x},i}(a_{\mathbf{x}+\hat{i},j}+a_{\mathbf{x}-\hat{j},j})\epsilon^{ij}}=e^{\frac{ik}{4\pi}\sum(\omega\cup
a-a\cup\omega)}\,,$ (29)

which, since $d\omega=0$, has the form of a framed Wilson loop. Specifically,
comparing to Eq. (21), $\widehat{W}_{q}(\tilde{C})=U\left[\frac{2\pi
q}{k}\star[\tilde{C}]\right]$. Indeed this operator implements the shift of
the gauge field

$U[\omega]^{\dagger}\,a_{\mathbf{x},i}\,U[\omega]=a_{x,i}-\frac{ik}{4\pi}\sum_{\mathbf{y},j,r}\omega_{\mathbf{y},j}\,\epsilon^{jr}\,[a_{\mathbf{y}+\hat{j},r}+a_{\mathbf{y}-\hat{r},r},a_{\mathbf{x},i}]=a_{\mathbf{x},i}+\omega_{\mathbf{x},i}\;,$
(30)

as is easily checked using Eq. (22). When $\omega=d\lambda$ is exact,
$U[d\lambda]$ implements a gauge transformation and must be equivalent to the
identity operator on the physical Hilbert space. Indeed, in this case

$U[d\lambda]=e^{\frac{ik}{4\pi}\sum(d\lambda\cup a-a\cup d\lambda)}\equiv
G[\lambda]=e^{-\frac{ik}{4\pi}\sum(\lambda\cup da+da\cup\lambda)}=1$ (31)

thanks to the Gauss law (8). A direct consequence is that a closed,
contractible framed Wilson loop is equivalent to a generator of a gauge
transformation, and hence is trivial. To see this, let $\lambda=\frac{2\pi
q}{k}\star[\tilde{D}]$, so that $d\lambda=\frac{2\pi
q}{k}\star[\partial\tilde{D}]$ where $\tilde{D}$ is a disk, and write

$G\left[\frac{2\pi
q}{k}\star[\tilde{D}]\right]=e^{-\frac{iq}{2}\sum(\star[\tilde{D}]\cup da-
da\cup\star[\tilde{D}])}=e^{\frac{iq}{2}\sum(\star[\partial\tilde{D}]\cup
a-a\cup\star[\partial\tilde{D}])}=\widehat{W}_{q}(\partial\tilde{D})\,,$ (32)

where we used summation by parts. Therefore, contractible framed Wilson loops
are trivial in the Hilbert space where the Gauss law is satisfied.

The generators of the 1-form symmetry (which again are just framed Wilson
loops) are indeed topological, since

$U[\omega]=G[\lambda]U[\omega]=e^{-\frac{2\pi i}{2k}\frac{1}{2}\sum
d\lambda\cup\omega-\omega\cup
d\lambda}\,U[\omega+d\lambda]=U[\omega+d\lambda]\,,$ (33)

where in the first equality we used the Gauss law (31), the second equality
follows from Eq. (25), and in the last equality we integrated by parts and
used $d\omega=0$. In terms of the Wilson loop, this implies

$\widehat{W}_{q}(\tilde{C})=G\left[\frac{2\pi
q}{k}\star[\tilde{D}]\right]\widehat{W}_{q}(\tilde{C})=\widehat{W}_{q}(\tilde{C}+\partial\tilde{D}).$
(34)

Intuitively, we can deform a framed Wilson loop by ‘tacking on’ an arbitrary
framed contractible loop, which is trivial thanks to the Gauss law. So Wilson
loops only depend on their homology class.

Now we consider another class of transformations which leaves the action
invariant and is associated with the staggered modes
$\mathcal{P}_{\text{staggered}}$. For this purpose, we take at least one of
$L_{1},L_{2}$ to be even. Consider a generic shift $a\rightarrow a+\omega$
with $\omega$ time-independent but not necessarily closed. The action changes
by

$\Delta S=-\frac{k}{4\pi}\int
dt\sum_{\Lambda}\left(da_{0}\cup\omega-\omega\cup da_{0}\right)\;.$ (35)

We can choose $\omega$ to be a staggered in the following sense. Consider a
fixed lattice site $x_{0}$ and take
$\omega_{\mathbf{x}_{0}+n\mathbf{s},i}=(-1)^{n}\,\varepsilon_{i}$ and all
other $\omega_{\mathbf{x},i}=0$, where $n\in\mathbb{Z}$ and
$\varepsilon_{1},\varepsilon_{2}\in\mathbb{R}$. This shift is consistent with
periodic boundary conditions as long as at least one of $L_{1},L_{2}$ is even.
It is easy to see that $X\cup\omega-\omega\cup X=0$ for _any_ 1-cochain $X$.
Explicitly, the transformation is given by

$a_{\mathbf{x},i}\rightarrow
a_{\mathbf{x},i}+\varepsilon_{i}\,\sum_{n=0}^{\text{lcm}(L_{1},L_{2})-1}(-1)^{n}\,\delta_{\mathbf{x},\mathbf{x}_{0}+n\mathbf{s}}\;,$
(36)

so that in Fourier components we have

$\begin{split}\tilde{a}_{\mathbf{p},i}&\rightarrow\tilde{a}_{\mathbf{p},i}+\varepsilon_{i}\,\frac{1}{\sqrt{V}}\sum_{n=0}^{\text{lcm}(L_{1},L_{2})-1}e^{in(p_{1}+p_{2}-\pi)}e^{i\mathbf{p}\cdot\mathbf{x}_{0}}\\\
&=\tilde{a}_{\mathbf{p},i}+\varepsilon_{i}\,e^{i\mathbf{p}\cdot\mathbf{x}_{0}}\frac{\text{lcm}(L_{1},L_{2})}{\sqrt{V}}\sum_{w\in\mathbb{Z}}\delta_{p_{1}+p_{2},\pi+2\pi
w}\end{split}$ (37)

The sum over $n$ forces the transformation to vanish unless
$\mathbf{p}\in\mathcal{P}_{\text{staggered}}$. In other words, this symmetry
only affects the zero modes discussed before. This transformation is
implemented by the exponentiated canonical momentum of
$\tilde{a}_{\mathbf{p},i}$, with
$\mathbf{p}\in\mathcal{P}_{\text{staggered}}$. But recall that any such
canonical momentum vanishes (15), and hence all states in the Hilbert space
are automatically invariant under the above transformation. In other words,
the above transformation is the trivial identity operator on the Hilbert
space. So the unusual staggered symmetry of the action is actually a gauge
redundancy, not a global symmetry. In particular any operator which is not
invariant under this symmetry (i.e. a generic unframed Wilson loop) would take
a state out of the Hilbert space and is therefore forbidden. The canonical
formulation makes it clear that symmetry of the lattice action that leads to
the projecting out of unframed Wilson lines is in fact a gauge redundancy, as
suggested in [9].

##  3 Compact Chern-Simons theory

We now want to construct the compact Chern-Simons theory with gauge group
$U(1)$. This is achieved by gauging the $2\pi\mathbb{Z}$ subgroup of the
$\mathbb{R}$ 1-form symmetry of the non-compact theory. Intuitively, this
turns the spatial gauge field into an angle-valued variable. For this purpose
we introduce integer plaquette field $n_{p}$, and write a continuous-time
action

$S=\frac{k}{4\pi}\int dt\sum_{\Lambda}\left[a\cup\dot{a}-a_{0}\cup(da-2\pi
n)-(da-2\pi n)\cup a_{0}\right]\,.$ (38)

Physically, $n_{p}$ can be thought of as the local magnetic flux, which can
change by an exact integer under a large gauge transformation — the sum of
$n_{p}$ over all of space is gauge-invariant. We demand that $a\rightarrow
a+2\pi m,n\rightarrow n+dm,m\in C^{1}(\mathbb{Z})$ is a gauge symmetry (with
$m$ time independent), as we will check in a moment. But first we check the
ordinary (small) gauge transformation $a\rightarrow a+d\lambda$ and
$a_{0}\rightarrow a_{0}+\dot{\lambda}$ are respected. Compared to the non-
compact case (7), we encounter additional terms

$\Delta S=\frac{k}{2}\int dt\sum_{\Lambda}n\cup\dot{\lambda}+\dot{\lambda}\cup
n\,.$ (39)

Since we take time to be non-compact we can integrate by parts in time and in
order for the action to be invariant we must set $\dot{n}=0$. This is quite
natural for an integer-valued field when time is continuous. Physically, this
means that there are no monopole events which would change the quantized
magnetic flux through space. To implement this constraint we follow the
modified Villain procedure and introduce a Lagrange multiplier field $\varphi$
on the sites of the lattice, and add the term

$\int dt\sum_{\Lambda}\,\dot{\varphi}\cup n$ (40)

to the action. When we quantize the theory, we impose that $\varphi$ is a
periodic scalar $\varphi\sim\varphi+2\pi$ so that its conjugate momentum, $n$,
becomes an integer-spectrum operator. The operator $e^{i\varphi}$ is nothing
but the monopole operator in our 2+1d $U(1)$ gauge theory.

Now it is also possible to cancel the above gauge variation using a shift of
$\varphi$. But this shift would involve a nonlocal shift of $\varphi$ by
$\lambda$ at nearby sites. Instead we proceed as in Ref.[9] and assign
$\varphi$ charge $k$ under the gauge symmetry $\varphi\to\varphi-k\lambda$,
while introducing an additional term

$\frac{k}{2}\int dt\sum_{\Lambda}\,\dot{a}\cup_{1}n\,.$ (41)

With these additional terms the variation of the action under small gauge
transformations is

$\Delta S=\frac{k}{2}\int dt\sum_{\Lambda}n\cup\dot{\lambda}-\dot{\lambda}\cup
n+d\dot{\lambda}\cup_{1}n=0$ (42)

using Eq. (2). Now let us verify that $a\rightarrow a+2\pi m$ and
$n\rightarrow n+dm$ (with $m$ time independent) is a symmetry:

$\Delta S=\frac{k}{4\pi}\int dt\sum_{\Lambda}2\pi m\cup\dot{a}+\int
dt\sum_{\Lambda}\dot{\varphi}\cup dm\,.$ (43)

Both terms are total time derivatives, so the action is invariant. To
summarize, the action

$S=\int dt\sum_{\Lambda}\frac{k}{4\pi}\left[a\cup\dot{a}-a_{0}\cup(da-2\pi
n)-(da-2\pi n)\cup a_{0}+2\pi\dot{a}\cup_{1}n\right]+\dot{\varphi}\cup n$ (44)

is invariant under the following transformations

$\begin{split}&a\rightarrow a+d\lambda+2\pi m\,\\\ &n\rightarrow n+dm\,,\\\
&\varphi\rightarrow\varphi-k\lambda\,,\end{split}$ (45)

where $\lambda\in C^{0}(\mathbb{R})$ and $m\in C^{1}(\mathbb{Z})$ and
$\dot{m}=0$. In fact, in the above discussion of the classical action we never
used the fact that $m$ was integer valued, and correspondingly $k$ could still
take any real value. This is due to the fact that we are taking time to be
non-compact, so that gauge transformations cannot wind in time. As we will see
below, even with time taken to be non-compact, level quantization and the
discreteness of large gauge transformations emerge once we quantize the
theory.

###  3.1 Quantization

In conventional lattice notation, the full action in continuum time is given
by

$S=\int
dt\sum_{\mathbf{x}}\Bigg{[}\frac{k}{4\pi}\left(a_{\mathbf{x},i}\dot{a}_{\mathbf{x}+\hat{i},j}\epsilon^{ij}-\left[(a_{0})_{\mathbf{x}}+(a_{0})_{\mathbf{x}+\mathbf{s}}\right]\left[(da)_{\mathbf{x},12}-2\pi
n_{\mathbf{x},12}\right]\right)\\\
+\left(\dot{\varphi}_{\mathbf{x}}-\frac{k}{2}(\dot{a}_{\mathbf{x},2}+\dot{a}_{\mathbf{x}+\hat{2},1})\right)n_{\mathbf{x},12}\Bigg{]}$
(46)

Again there is no dependence on $\dot{a}_{0}$, so the canonical momentum of
$a_{0}$ vanishes, which in turn implies the Gauss law (the normalization is
for later convenience)

$\mathcal{G}_{\mathbf{x}}=\frac{k}{4\pi}\left[(da-2\pi
n)_{\mathbf{x}-\mathbf{s},12}+(da-2\pi n)_{\mathbf{x},12}\right]=0\;.$ (47)

Switching to momentum space, i.e.

$\displaystyle
a_{\mathbf{x},i}=\frac{1}{\sqrt{V}}\sum_{\mathbf{p}}\tilde{a}_{\mathbf{p},i}\,e^{i\mathbf{p}\cdot\mathbf{x}}\;,\
n_{\mathbf{x},12}=\frac{1}{\sqrt{V}}\sum_{\mathbf{p}}\tilde{n}_{\mathbf{p}}\,e^{i\mathbf{p}\cdot\mathbf{x}}\;,\
\varphi_{\mathbf{x}}=\frac{1}{\sqrt{V}}\sum_{\mathbf{p}}\tilde{\varphi}_{\mathbf{p}}\,e^{i\mathbf{p}\cdot\mathbf{x}}\;,$
(48)

we get that the rest of the action can be written as

$S=\sum_{\mathbf{p}}\left(\frac{1}{2}K(\mathbf{p})^{ij}\,\tilde{a}_{\mathbf{p},i}\,\dot{\tilde{a}}_{-\mathbf{p},j}+\tilde{n}_{\mathbf{p}}\left(\dot{\tilde{\varphi}}_{-\mathbf{p}}-\frac{k}{2}\dot{\tilde{a}}_{-\mathbf{p},2}-\frac{k}{2}\dot{\tilde{a}}_{-\mathbf{p},1}\,e^{-ip_{2}}\right)\right)\;.$
(49)

Defining
$\overline{\tilde{\varphi}}_{\mathbf{p}}=\tilde{\varphi}_{\mathbf{p}}-\frac{k}{2}\tilde{a}_{\mathbf{p},2}-\frac{k}{2}\tilde{a}_{\mathbf{p},1}\,e^{ip_{2}}$
we write the above as

$S=\sum_{\mathbf{p}}\left(\frac{1}{2}K(\mathbf{p})^{ij}\,\tilde{a}_{\mathbf{p},i}\dot{\tilde{a}}_{-\mathbf{p},j}+\tilde{n}_{\mathbf{p}}\dot{\overline{\tilde{\varphi}}}_{-\mathbf{p}}\right)\;.$
(50)

This form of the action makes it clear that when
$\mathbf{p}\not\in\mathcal{P}_{\text{staggered}}$ the operators
$\tilde{a}_{\mathbf{p},i}$ obey the commutation relations (17) as before, and
the canonical momenta $\Pi_{p,i}$ of modes $\tilde{a}_{\mathbf{p},i}$ with
$\mathbf{p}\in\mathcal{P}_{\text{staggered}}$ all vanish. On the other hand we
have that

$[\overline{\tilde{\varphi}}_{\mathbf{p}},\tilde{n}_{\mathbf{q}}]=i\delta_{\mathbf{p},-\mathbf{q}}\,.$
(51)

Now we assume that $\tilde{n}_{\mathbf{p}}$ commutes with
$\tilde{a}_{\mathbf{p},i}$ to get

$[\tilde{\varphi}_{\mathbf{p}},\tilde{n}_{\mathbf{q}}]=i\delta_{\mathbf{p},-\mathbf{q}}\
\Longrightarrow\
[\varphi_{\mathbf{x}},n_{\mathbf{y},12}]=i\delta_{\mathbf{x},\mathbf{y}}\,.$
(52)

So we can view $n$ as the canonical momentum for $\varphi$. But now notice
that since

$[\tilde{a}_{\mathbf{p},i},\overline{\tilde{\varphi}}_{\mathbf{q}}]=0=[\tilde{a}_{\mathbf{p},i},\tilde{\varphi}_{\mathbf{q}}-\frac{k}{2}\tilde{a}_{\mathbf{q},2}-\frac{k}{2}\tilde{a}_{\mathbf{q},1}\,e^{iq_{2}}]\,,$
(53)

$\tilde{\varphi}$ will have a non-trivial commutation relation with the gauge
field (for $\mathbf{p}\not\in\mathcal{P}_{\text{staggered}}$)

$[\tilde{\varphi}_{\mathbf{p}},\tilde{a}_{\mathbf{q},i}]=i\pi\,\delta_{\mathbf{p},-\mathbf{q}}\frac{1+e^{i(p_{1}+p_{2})}}{1+\cos(p_{1}+p_{2})}(\delta_{i1}\,e^{-ip_{1}}-\delta_{i2})\,.$
(54)

For instance, in position space this implies that

$[\varphi_{\mathbf{x}},a_{\mathbf{y},i}+a_{\mathbf{y}+\mathbf{s},i}]=2\pi
i\,(\delta_{i1}\delta_{\mathbf{x},\mathbf{y}+\hat{1}}-\delta_{i2}\delta_{\mathbf{x},\mathbf{y}})\,,$
(55)

so that

$[\varphi_{\mathbf{x}},(da)_{\mathbf{y},12}+(da)_{\mathbf{y}+\mathbf{s},12}]=2\pi
i\,(\delta_{\mathbf{x},\mathbf{y}}-\delta_{\mathbf{x},\mathbf{y}+\mathbf{s}})\,.$
(56)

This leads to
$[\varphi_{\mathbf{x}},\mathcal{G}_{\mathbf{y}}]=-ik\,\delta_{\mathbf{x},\mathbf{y}}$,
so indeed the monopole operator
$\mathcal{M}_{\mathbf{x}}=e^{i\varphi_{\mathbf{x}}}$ has electric charge $-k$
[36, 37, 9]. In the following, we will also make use of the fact that

$\sum_{\Lambda}\left(\varphi\cup X+\frac{k}{2}a\cup_{1}X\right)$ (57)

commutes with all $a$, where here $X$ is an arbitrary 2-cochain.

###  3.2 Gauge redundancies

Conventional 0-form gauge transformations $a\rightarrow a+d\lambda$,
$\varphi\to\varphi-k\lambda$ are implemented by the analogous operator to
(31), namely

$G[\lambda]=e^{-\frac{ik}{4\pi}\sum\lambda\cup(da-2\pi n)+(da-2\pi
n)\cup\lambda}\,,$ (58)

which is equal to the identity operator on Hilbert space thanks to the Gauss
law (47). An important consequence of the Gauss law is that the total magnetic
flux through the torus necessarily vanishes.

We also want to impose compactness of $\varphi$, in other words we want to
gauge discrete shifts $\varphi\to\varphi+2\pi u$ with $u$ an independent
integer on each site. This transformation is achieved by

$C[u]=e^{2\pi i\sum u\cup n}\,,\quad u\in C^{0}(\mathbb{Z})\,.$ (59)

Setting this operator equal to the identity on the physical Hilbert space
implies that $n$ is an operator with an integer spectrum.

Finally, we have the discrete gauge symmetry corresponding to large gauge
transformations, which take $a\rightarrow a+2\pi m,n\rightarrow n+dm$ with
$m\in C^{1}(\mathbb{Z})$. This is implemented by the operator

$U[m]=e^{i\sum\left[\frac{k}{2}(m\cup a-a\cup m+a\cup_{1}dm)+\varphi\cup
dm\right]}\,,\quad m\in C^{1}(\mathbb{Z})\,.$ (60)

Now we must impose the constraint that $U[m]$ equals the identity operator on
Hilbert space. For this to be consistent we also have to demand that the
$U[m]$ operators are invariant under ordinary gauge transformations, but also
commute between themselves, i.e. are themselves invariant under large gauge
transformations. This requirement is what leads to the quantization of the
level $k$ It is easy to verify that $U[m]$ and $G[\lambda]$ commute. However,
in general the generators of large gauge transformations do not commute among
themselves, but instead have the following commutation relations:

$U[m]\,U[m^{\prime}]=U[m^{\prime}]\,U[m]\,e^{ik\pi\sum m\cup
m^{\prime}-m^{\prime}\cup m}\,.$ (61)

For generic values of $k$, this non-commutativity prevents us from imposing
the constraint $U[m]=1$ on the Hilbert space, and can be thought of as a ’t
Hooft anomaly for the $2\pi\mathbb{Z}$ subgroup of the 1-form symmetry in the
non-compact $\mathbb{R}_{k}$ Chern-Simons theory. Clearly if $k$ is an even
integer, the anomaly trivializes, and imposing the 1-form gauge constraint
defines the physical Hilbert space of the $U(1)_{k}$ theory.

It is well-known that when $k$ is an odd integer the $U(1)_{k}$ CS theory is a
spin-TQFT [38, 39]. In the current context, if $k$ is an odd integer one can
introduce fermionic degrees of freedom on the lattice to cancel the unwanted
phase in Eq. (61) in a procedure known as fermion condensation [25, 40, 41,
42]. For the moment we restrict ourselves to $k\in 2\mathbb{Z}$, and discuss
the fermionic CS theory in Sec. 4 below.

In fact, to consistently gauge $2\pi$ shifts $a\to a+2\pi m$ it is necessary
but not sufficient to ensure that the generators $U[m]$ commute. To see the
remaining issue, note that the operators as defined in Eq. (60) satisfy

$U[m]\,U[m^{\prime}]=U[m+m^{\prime}]\,e^{i\frac{k\pi}{2}\sum m\cup
m^{\prime}-m^{\prime}\cup m}\,.$ (62)

If $dm=dm^{\prime}=0$, this phase is indeed trivial. Otherwise this phase can
be nontrivial if $k$ is not a multiple of 4, and hence inconsistent with the
constraint $U[m]=1$. We can fix this problem by rephasing the operators by
$U[m]\to e^{i\frac{k\pi}{2}\sum m\cup m}\,U[m]$ so that

$U[m]\,U[m^{\prime}]=U[m+m^{\prime}]\,e^{ik\pi\sum m\cup m^{\prime}}\,,$ (63)

which is innocuous if $k\in 2\mathbb{Z}$.

###  3.3 $\mathbb{Z}_{k}$ 1-form symmetry and ’t Hooft anomaly

Recall that in the non-compact CS theory, framed Wilson loops with charge $q$
consisted of two charge-$q/2$ Wilson loops displaced by $\mathbf{s}$. In the
compact CS theory, when $q$ is odd the two Wilson loops have fractional
charges and have to be connected by a $\mathbb{Z}_{2}$ surface built out of
the Villain variable $n_{p}$,

$\widehat{W}_{q}(\tilde{C})=e^{\frac{iq}{2}\sum\star[\tilde{C}]\cup
a-a\cup\star[\tilde{C}]+2\pi\star[\tilde{C}]\cup_{1}n}\,.$ (64)

Throughout the following we denote the minimally-charged Wilson line (which is
now a well-defined notion) by
$\widehat{W}(\tilde{C})\equiv\widehat{W}_{1}(\tilde{C})$. We can repeat the
analysis in Sec. 2.1 to show that these framed spatial Wilson loops only
depend on the homology class of $\tilde{C}$. To see this, let us use the 2d
cup product identity from Footnote 4

$\lambda\cup n-n\cup\lambda=d\lambda\cup_{1}n,$ (65)

to rewrite the generator of gauge transformations (58) as

$G[\lambda]=e^{\frac{ik}{4\pi}\sum(d\lambda\cup a-a\cup d\lambda)+4\pi
n\cup\lambda+2\pi\,d\lambda\cup_{1}n}\,.$ (66)

Now we set $\lambda=\frac{2\pi q}{k}\star[\tilde{D}]$, so that
$d\lambda=\frac{2\pi q}{k}\star[\partial\tilde{D}]$ where $\tilde{D}$ is a
disk and $q\in\mathbb{Z}$. Then one finds

$G\left[\frac{2\pi
q}{k}\star[\tilde{D}]\right]=e^{\frac{iq}{2}\sum(\star[\partial\tilde{D}]\cup
a-a\cup\star[\partial\tilde{D}])+4\pi
n\cup\star[\tilde{D}]+2\pi\star[\partial\tilde{D}]\cup_{1}n}=\widehat{W}_{q}(\partial\tilde{D})\,,$
(67)

so that contractible framed Wilson loops are pure gauge transformations and
hence trivial, provided the Gauss law is satisfied. To see very explicitly
that framed Wilson loops are topological, we can start with an arbitrary loop
$\tilde{C}$ and add a contractible loop $\partial\tilde{D}$ using a gauge
transformation

$\widehat{W}(\tilde{C})=G\left[\frac{2\pi}{k}\star[\tilde{D}]\right]\,\widehat{W}(\tilde{C})=\widehat{W}(\tilde{C}+\partial\tilde{D})\,,$
(68)

just as in the non-compact case.

While contractible Wilson loops can be trivialized using a gauge
transformation, non-contractible loops are necessarily non-trivial as they
obey the familiar commutation relation

$\widehat{W}(\tilde{C}_{1})\widehat{W}(\tilde{C}_{2})=\widehat{W}(\tilde{C}_{2})\widehat{W}(\tilde{C}_{1})=e^{-\frac{2\pi
i}{k}\text{Int}(\tilde{C}_{1},\tilde{C}_{2})}\,.$ (69)

Moreover, in contrast to the non-compact case, we have that

$[\widehat{W}(\tilde{C})]^{k}=\widehat{W}_{k}(\tilde{C})=e^{\frac{ik}{2}\sum\star[\tilde{C}]\cup
a-a\cup\star[\tilde{C}]+2\pi\star[\tilde{C}]\cup_{1}n}=U[\star[\tilde{C}]]=1$
(70)

by the large-gauge constraint (we remind the reader that in the current
discussion we are taking $k$ to be even). So while contractible Wilson loops
are trivial, the eigenvalues of the minimal non-contractible Wilson loops are
non-trivial $k$th roots of unity.

In particular, just as in the continuum, these relations imply the $k$-fold
degeneracy of the Hilbert space on the torus. Of course, this commutation
relation can be viewed as a central extension of the
$\mathbb{Z}_{k}\times\mathbb{Z}_{k}$ symmetry acting on the Wilson loops
winding around each cycle of the torus — the hallmark of the ’t Hooft anomaly
of the $\mathbb{Z}_{k}^{(1)}$ 1-form symmetry generated by the Wilson loops
themselves.

We emphasize that although the ingredients of our lattice model are bosonic,
each carrying an infinite dimensional Hilbert space, the actual Hilbert space
of the theory is finite and $k$-dimensional on the torus. While the non-
trivial algebra obeyed by Wilson loops implies that all states in the Hilbert
space are _at least_ $k$-fold degenerate, the fact that the only non-trivial
operators in the theory are the topological framed Wilson loops means that the
Hilbert space is finite dimensional. Just as in the continuum, the finiteness
of the Hilbert space is spoiled if we include a conventional lattice Maxwell
term, which makes the Hamiltonian non-zero and reinstates the unframed Wilson
lines as physical and non-topological operators.

###  3.4 Defect Hilbert spaces and topological spin

Up until this point, we have always been working in the Hilbert space where
the local Gauss law (47) and the framing constraint are both satisfied. While
this gives rise to a well-defined Hilbert space, we can also access other
superselection sectors where the constraints are violated in a controlled way.
Such modifications of the contraints give rise to ‘defect’ Hilbert spaces.

In the case of the Gauss law, the defect Hilbert space corresponds to
inserting temporal Wilson lines which represent the static worldlines of
electric probe particles. If at least one of $L_{1},L_{2}$ is even, the
Lagrangian in Eq. (44) is invariant under staggered shifts
$(a_{0})_{\mathbf{x}}\to(a_{0})_{\mathbf{x}}+\omega\sum_{n=0}^{\text{lcm}(L_{1},L_{2})-1}(-1)^{n}\delta_{\mathbf{x},\mathbf{x}_{0}+n\mathbf{s}}$,
suggesting that we should only consider _framed_ temporal Wilson lines.
Indeed, suppose we insert an ordinary, unframed temporal Wilson line of charge
$q$ at a site at $\mathbf{y}$. This modifies the Gauss law to

$(da-2\pi n)_{\mathbf{x}-\mathbf{s},12}+(da-2\pi
n)_{\mathbf{x},12}\stackrel{{\scriptstyle?}}{{=}}\frac{4\pi
q}{k}\delta_{\mathbf{x},\mathbf{y}}\,.$ (71)

This constraint would be inconsistent — in momentum space, the left-hand-side
does not involve the $\mathcal{P}_{\text{staggered}}$ modes, while the right-
hand-side does. To get a sensible defect, in general we have to split a
charge-$q$ temporal Wilson line into a staggered pair of charge-$q/2$ lines,
which gives rise to the modified Gauss law

$(da-2\pi n)_{\mathbf{x}-\mathbf{s},12}+(da-2\pi n)_{\mathbf{x},12}=\frac{2\pi
q}{k}(\delta_{\mathbf{x},\tilde{\mathbf{x}}_{0}-\frac{\mathbf{s}}{2}}+\delta_{\mathbf{x},\tilde{\mathbf{x}}_{0}+\frac{\mathbf{s}}{2}})\,,$
(72)

where $\tilde{\mathbf{x}}_{0}$ is a point on the dual lattice corresponding to
the midpoint of the two charge-$q/2$ probes.

The most general defect Hilbert space, which we denote by
$\mathcal{H}_{\\{(q_{i},\tilde{\mathbf{x}}_{i})\\}}$ is labelled by the
charges $q$ and positions $\tilde{\mathbf{x}}$ of the static probe particles.
The defects are topological, corresponding to the fact that the framed
temporal Wilson lines are nothing but the defects associated to the
$\mathbb{Z}_{k}$ 1-form symmetry. Explicitly, given a state
$|\psi\rangle\in\mathcal{H}_{(q,\tilde{\mathbf{x}})}$ we have the unitary
transformation

$|\psi\rangle\to\widehat{W}_{q}(\tilde{C}_{\tilde{\mathbf{x}}\tilde{\mathbf{y}}})|\psi\rangle\,,$
(73)

which maps
$\mathcal{H}_{(q,\tilde{\mathbf{x}})}\to\mathcal{H}_{(q,\tilde{\mathbf{y}})}$.
Here $\tilde{C}_{\tilde{\mathbf{x}},\tilde{\mathbf{y}}}$ is any curve on the
dual lattice connecting $\tilde{\mathbf{x}}$ and $\tilde{\mathbf{y}}$. The
dependence on this curve is also topological, up to phases which can appear if
the curve crosses other defects — this is related to the anomaly. In general,
we can connect two defect Hilbert spaces
$\mathcal{H}_{\\{(q_{i},\tilde{\mathbf{x}}_{i})\\}}$ and
$\mathcal{H}_{\\{(q_{i}^{\prime},\tilde{\mathbf{x}}_{i}^{\prime})\\}}$ if and
only if $(\sum q_{i})=(\sum q_{i}^{\prime})\text{ mod }k$, so that they have
the same charge under the 1-form symmetry. Indeed, the defect Hilbert space
with a charge-$k$ temporal Wilson line at $\tilde{\mathbf{x}}$ can be mapped
to the ordinary Hilbert space by acting with the local monopole operator
$e^{-i\bar{\varphi}_{\tilde{\mathbf{x}}-\frac{\mathbf{s}}{2}}}$, reflecting
the fact that the 1-form symmetry is $\mathbb{Z}_{k}$.

We can now compute a correlation function which we interpret as giving the
topological spin. We consider the expectation value of a framed Wilson loop on
a contractible curve $\tilde{C}$, but in the defect Hilbert space
$\mathcal{H}_{(1,\tilde{\mathbf{x}})}$ where $\tilde{\mathbf{x}}$ and
$\tilde{C}$ are depicted in Fig. 5. We can use the argument near Eq. (68)
above to deform the Wilson loop to a single (staggered) plaquette,

$\langle\widehat{W}(\tilde{C})\rangle_{(1,\tilde{\mathbf{x}})}=\langle
e^{\frac{i}{2}\left((da-2\pi
n)_{\tilde{\mathbf{x}}-\frac{3\mathbf{s}}{2}}+(da-2\pi
n)_{\tilde{\mathbf{x}}-\frac{\mathbf{s}}{2}}\right)}\rangle_{(1,\tilde{\mathbf{x}})}=e^{\frac{2\pi
i}{2k}}$ (74)

using the modified Gauss constraint (72). Alternatively, we can arrive at the
same phase by applying an open spatial Wilson line to move the defect away
from the loop $\tilde{C}$, shrinking the loop, and moving the defect back to
its original location.

Figure 5:  Shrinking a contractible Wilson loop in the presence of a temporal
Wilson line yields a $\mathbb{Z}_{2k}$ phase which we interpret as the
topological spin. The pair of circular blue dots denote the two charge-$1/2$
edges of the framed temporal Wilson line.

We can also consider a defect Hilbert space corresponding to a fixed violation
of the staggered constraint, for instance by setting
$\Pi_{\mathbf{p},i}=c_{\mathbf{p},i}$ where $c_{\mathbf{p},i}$ are some
c-numbers and $\mathbf{p}\in\mathcal{P}_{\text{staggered}}$. Note that this
does not define a _local_ defect Hilbert space, as the constraint is modified
locally in momentum space. These defect Hilbert spaces correspond to
superselection sectors with fixed eigenvalues under (linear combinations of)
the transformations discussed near Eq. (36), and are mixed by acting with
ordinary, unframed spatial Wilson loops. Breaking the staggered gauge
redundancy explicitly by turning on a Maxwell term, or coupling to an
electrically charged particle with a standard hopping term, will lead to
dynamical mixing of these superselection sectors, and result in an infinite-
dimensional Hilbert space, as discussed in Sec. 3.3.

##  4 Odd levels: the fermionic Chern-Simons theory

In Sec. 3.2 we observed that when $k$ is odd the (naive) generators of large
gauge transformations do not commute, but instead satisfy

$U[m]\,U[m^{\prime}]=U[m+m^{\prime}]\,e^{ik\pi\sum m\cup
m^{\prime}}=U[m^{\prime}]\,U[m]\,e^{ik\pi\sum m\cup m^{\prime}+m^{\prime}\cup
m}\,.$ (75)

These non-trivial phases constitute an obstruction to setting $U[m]=1$ to
define the physical gauge-invariant Hilbert space, i.e. a ’t Hooft anomaly for
the $2\pi\mathbb{Z}$ subgroup of the $\mathbb{R}$ 1-form symmetry of the
$\mathbb{R}_{k}$ theory.777Alternatively, one can start with $\mathbb{R}_{k}$
and gauge the non-anomalous $4\pi\mathbb{Z}$ subgroup of the 1-form symmetry
to land on the $U(1)_{4k}$ theory. The $\mathbb{Z}_{2}$ subgroup of the
$\mathbb{Z}_{4k}$ 1-form symmetry of $U(1)_{4k}$ has an ’t Hooft anomaly, with
the generators obeying (75).

An intuitive explanation for why there is an anomaly is as follows. Via
Poincaré duality we can view the generators of large gauge transformations as
operators supported on (not necessarily closed) lines. In fact the operator
$U[m]$ is just a charge-$k$ framed Wilson loop, but _without_ the surface
connecting the two edges of the ribbon (the last term in Eq. (64)). When $k$
is odd, the Wilson lines at the edges of the ribbon have fractional charge and
are not invariant under large gauge transformations, leading to the anomaly.
One can attempt to fix this by including Villain fields in the interior of the
ribbon, but the resulting operator acts on $\varphi$ and is therefore not the
correct generator of large gauge transformations.

We can cancel the anomaly by finding a suitable set of operators $S[m]$ which
satisfy the same relations as Eq. (75), so that we can define an improved
generator of large gauge transformations $U[m]\,S[m]$ which can be
consistently constrained to the identity on Hilbert space. This is only
possible if we first enlarge our Hilbert space and the algebra of operators.
There is not a unique choice for these additional degrees of freedom — we will
follow Ref. [26] and build $S[m]$ out of a set of fermionic operators. The
values of $S[m]$ on cohomology classes $m\in H^{1}(\mathbb{Z})$ turn out to be
related to the spin structure on the spatial torus. This is how the odd $k$
theory, which is a spin-TQFT, depends on spin structure. The same operators we
discuss below were used in Refs. [43, 44, 45] to define generalized Jordan-
Wigner bosonization maps on the lattice.

To proceed, we introduce a pair of Majorana fermions
$\gamma_{p},\gamma_{p}^{\prime}$ on plaquettes, with the usual algebra

$\\{\gamma_{p},\gamma_{p^{\prime}}\\}=\\{\gamma_{p}^{\prime},\gamma_{p^{\prime}}^{\prime}\\}=2\delta_{p,p^{\prime}}\,,\
\\{\gamma_{p},\gamma_{p^{\prime}}^{\prime}\\}=0\,.$ (76)

The full Hilbert space is now enlarged to include the $2^{N}$-dimensional
fermionic Hilbert space — in a moment we will project down to a subspace which
has the same dimension of the bosonic Hilbert space we started with. One can
think of these Majoranas as the real and imaginary parts of the creation and
annihilation operators

$c_{p}=\frac{1}{2}(\gamma_{p}+i\gamma_{p}^{\prime}),\
c_{p}^{\dagger}=\frac{1}{2}(\gamma_{p}-i\gamma_{p}^{\prime})\,.$ (77)

We define fermion parity on each plaquette to be
$(-1)^{F_{p}}=i\gamma_{p}^{\prime}\gamma_{p}$.

The crucial building blocks for defining the fermionic CS theory are a set of
link operators, schematically defined as
$S_{\ell}=i\gamma_{p}\gamma_{p^{\prime}}^{\prime}$ where $p$ and $p^{\prime}$
are the (positively oriented) plaquettes lying in the coboundary of $\ell$.
More specifically, these operators are defined for positively-oriented links,

$S_{\mathbf{x},1}=i\gamma_{\mathbf{x},12}\,\gamma^{\prime}_{\mathbf{x}-\hat{2},12}\,,\quad
S_{\mathbf{x},2}=i\gamma_{\mathbf{x}-\hat{1},12}\,\gamma^{\prime}_{\mathbf{x},12}\,,$
(78)

and extended to negatively oriented links via $S_{-\ell}=-S_{\ell}$, just as
for a 1-cochain. These operators square to 1 and satisfy [26, 43, 45]

$S_{\ell}S_{\ell^{\prime}}=S_{\ell^{\prime}}S_{\ell}\,e^{i\pi\sum[\ell]\cup[\ell^{\prime}]+[\ell^{\prime}]\cup[\ell]}\,.$
(79)

Now let $m\in C^{1}(\mathbb{Z})$ and define [26]

$S[m]=e^{i\pi\sum_{\ell<\ell^{\prime}\in\lambda}[\ell]\cup[\ell^{\prime}]}\prod_{\ell\in\lambda}S_{\ell}\,,$
(80)

where $\lambda$ is the ordered set of all links $\ell$ for which
$m_{\ell}\not=0$ mod $2$, and the product is taken from left to right in the
order dictated by $\lambda$. The ordering-dependence of the two factors
compensate so that the operator as a whole is independent of the choice of
ordering. If $m=dq$ is exact, then

$S[dq]=\prod_{p}(i\gamma_{p}^{\prime}\gamma_{p})^{dq\cup_{1}[p]}\equiv
e^{i\pi\sum dq\cup_{1}F}$ (81)

where $F_{p}$ is the fermion number (well-defined mod 2) of a given plaquette.
If $dm=0$ but is not exact, then $S[m]=-e^{i\pi\sum m\cup_{1}F}$. Finally, if
$dm\not=0$ mod $2$, then

$S[m]=e^{i\pi\sum m\cup_{1}F}\prod_{p}\gamma_{p}^{|(dm)_{p}|}$ (82)

with some fixed ordering of the product over $\gamma$’s. Note that these
operators satisfy [26]

$S[m]\,S[m^{\prime}]=S[m+m^{\prime}]\,e^{i\pi\sum m\cup
m^{\prime}}=S[m^{\prime}]\,S[m]\,e^{i\pi\sum m\cup m^{\prime}+m^{\prime}\cup
m}\,,$ (83)

reproducing the algebra Eq. (75).

To see how the spin structure comes into play, note that we are free to
consider a family of operators

$S_{\Gamma}[m]=e^{i\pi\sum_{\Gamma}m}\,S[m]\,,$ (84)

where the additional phase factor depends on a closed curve $\Gamma$ on the
lattice, $\partial\Gamma=0$. This phase factor preserves the algebra (83), and
if $dm=0$, the operator above is just a sign times $e^{i\pi\sum m\cup_{1}F}$.
Let $m=\star[\tilde{C}]$ with $\partial\tilde{C}=0$. Specifically, we find

$S_{\Gamma}[\star[\tilde{C}]]=e^{i\pi\sum\star[\tilde{C}]\cup_{1}F}\begin{cases}(+1)\,e^{i\pi\,\text{Int}(\Gamma,\,\tilde{C})}=+1&\text{
if $\tilde{C}$ is contractible,}\\\
(-1)\,e^{i\pi\,\text{Int}(\Gamma,\,\tilde{C})}&\text{ if $\tilde{C}$ is non-
contractible.}\end{cases}$ (85)

Now we would like to interpret these phases in terms of the spin structure on
the torus. Following e.g. Ref. [46, 47], given a spin structure $\rho$ on the
torus there is an associated quadratic form $q_{\rho}$ on
$H_{1}(\mathbb{Z}_{2})$ which is equal to 0 on cycles with anti-periodic
boundary conditions and 1 on cycles with periodic boundary conditions. This
form satisfies
$q_{\rho}(C_{1}+C_{2})=q_{\rho}(C_{1})+q_{\rho}(C_{2})+\text{Int}(C_{1},C_{2})$.
We can identify
$e^{i\pi\sum\star[\tilde{C}]\cup_{1}F}\,S_{\Gamma}[\star[\tilde{C}]]=e^{i\pi
q_{\rho}(\tilde{C})}$, where the choice of $\Gamma$ is equivalent to a choice
$\rho$ of spin structure. For this discussion only the $\mathbb{Z}_{2}$
homology of $\Gamma$ is important. Denoting the two cycles of the torus by
$a,b$, we find

$\rho$ | AP/AP | AP/P | P/AP | P/P
---|---|---|---|---
$\Gamma$ | $a+b$ | $b$ | $a$ | $0$
(86)

In short, the choice of $\Gamma$ is simply telling us which generating cycles
of the torus have anti-periodic boundary conditions. Finally, if $dm\not=0$,
the operator $S_{\Gamma}[m]$ is equal to $e^{i\pi\sum
m\cup_{1}F}\prod_{p}\gamma_{p}^{|(dm)_{p}|}$, up to a sign.

As described in the beginning of this section, the idea is to replace the
constraint $U[m]=1$, which is inconsistent when $k$ is odd, with
$U[m]\,S_{\Gamma}[m]=1$. When $dm=0$ mod $2$ this sets $U[m]$ equal to a sign
times $(-1)^{\sum m\cup_{1}F}$, but when $dm\not=0$ mod $2$ this equates the
open Wilson line (which acts on the bosonic Hilbert space) with a pair of
Majorana operators at the endpoints (which act on the fermionic Hilbert
space):

$U[m]=e^{i\frac{k\pi}{2}\sum m\cup m}\,e^{i\sum\frac{k}{2}(m\cup a-a\cup
m+a\cup_{1}dm)+\varphi\cup dm}=e^{i\pi\sum_{\Gamma}m}\,e^{i\pi\sum
m\cup_{1}F}\,\prod_{p}\gamma_{p}^{|(dm)_{p}|}\,.$ (87)

The right hand side changes the fermion number by one unit on the plaquettes
where $(dm)_{p}=1$ mod $2$, while the left hand side shifts $n\to n+dm$. For
this to be consistent, we must identify the fermion parity on a plaquette with
the eigenvalue of the Villain field mod $2$:

$(-1)^{n_{p}}=(-1)^{F_{p}}\,.$ (88)

The physical Hilbert space is defined by this spin-charge relation (88)
together with the large-gauge constraint

$U[m]\,S_{\Gamma}[m]=1\,.$ (89)

We stress that the Hilbert space is defined with reference to some fixed
$\Gamma$. Recall that in order to introduce the fermionic operators
$\gamma,\gamma^{\prime}$ to cancel the anomaly, we had to enlarge our Hilbert
space by a factor of $2^{N}$. Physically, however, we should have a
$k$-dimensional Hilbert space on the torus. The point is that the
identification (88) picks out just a single fermionic state for each bosonic
state in the large Hilbert space unconstrained by (89), so that the physical
Hilbert space has dimension $k$.

Now consider the implications of the above discussion for a charge-$k$ framed
Wilson loop. Using the large-gauge constraint, the spin-charge relation, and
Eq. (85), we find

$\begin{split}\widehat{W}_{k}(\tilde{C})=e^{ik\pi\sum\star[\tilde{C}]\cup_{1}n}\,U[\star[\tilde{C}]]&=e^{ik\pi\sum\star[\tilde{C}]\cup_{1}F}\,S_{\Gamma}[\star[\tilde{C}]]\\\
&=\begin{cases}+1&\text{ if $\tilde{C}$ is contractible,}\\\
(-1)\,e^{i\pi\,\text{Int}(\Gamma,\,\tilde{C})}&\text{ if $\tilde{C}$ is non-
contractible.}\end{cases}\end{split}$ (90)

For any choice of $\Gamma$, charge-$k$ contractible Wilson loops are always
equal to 1 (in the absence of defects), while winding charge-$k$ Wilson loops
are a non-trivial sign depending on the spin structure:

$\widehat{W}_{k}(\tilde{C})=(-1)\,e^{i\pi\,\text{Int}(\Gamma,\,\tilde{C})}\,.$
(91)

In particular, the value of a charge-$k$ framed Wilson loop wrapped on a non-
trivial cycle changes by a sign when we change the spin-structure. This
matches the continuum discussions in Refs. [48, 49].

Furthermore, the spin-charge relation tells us that the charge-$k$ temporal
Wilson line represents the worldline of a fermion — in the defect Hilbert
space we can sum the modified Gauss law (72) over the entire lattice to find
$\sum_{p}n_{p}=-1$, so that the net fermion parity is

$(-1)^{\sum_{p}F_{p}}=-1\,.$ (92)

The spin-charge relation also tells us that the monopole operator is a fermion
— $e^{i\varphi_{\mathbf{x}}}$ anti-commutes with $e^{i\pi n_{\mathbf{x},12}}$,
violating the spin-charge relation. This can be fixed by dressing the monopole
with a fermion, e.g.
$\gamma_{\mathbf{x},12}^{\prime}\,e^{i\varphi_{\mathbf{x}}}$. Of course, this
operator is not gauge-invariant, and in the pure CS theory monopole operators
have to be dressed by an open charge-$k$ Wilson line. In fact such an open
Wilson line is nothing but the (trivial) operator $U[m]S_{\Gamma}[m]$. We show
an example in Fig. 6.

Figure 6:  An open Wilson line equal to the identity operator by the large-
gauge constraint.

If we add electrically charged matter fields, we can define a genuine local
monopole operator which is necessarily a fermion when $k$ is odd. In the
continuum, such a fermionic monopole operator is nothing but the expected
image of the Dirac fermion $\Psi$ in the basic bosonization map relating
$\Psi$ to $U(1)_{1}$ coupled to a unit charge scalar [50, 28, 51].

##  5 Conclusions

In this work we have applied the Villain Hamiltonian approach to the canonical
quantization of $U(1)_{k}$ Chern-Simons theory on a spatial lattice. Our
discretization preserves defining properties of the continuum theory —
compactness of the gauge group, quantization of the level, the 1-form symmetry
and its ’t Hooft anomaly, the ground state degeneracy, and even its fermionic
nature when the level is odd. More importantly, the lattice construction
clarifies subtle aspects of the continuum theory. For instance, while in the
continuum monopole operators are disorder operators, within the modified
Villain approach on the lattice they have local expressions in terms of
fields. The electric charges of these monopole operators arises naturally in
our construction. Similarly, we identified how the framing of Wilson lines,
which in the continuum is quite subtle, is incorporated on the lattice through
constraints on the physical Hilbert space. As a result of this mechanism, the
physical operators of the lattice theory are topological ribbons, whose
algebra implies a $k$-fold degeneracy on the torus.

The Hilbert space of our model is defined by a local gauge constraint (Gauss
law), discrete gauge constraints imposing compactness of the $U(1)$ gauge
group and of the (phase of the) monopole operator $\varphi$, and (depending on
the details of the spatial torus) non-local constraints projecting out
unframed Wilson loops. These constraints constitute a collection of ‘commuting
projectors’ which can in principle be used to define commuting projector
Hamiltonians [52, 53] where the various constraints are energetically imposed.
In particular, when the number of sites in each direction is odd the non-local
constraints disappear (since in that case ordinary Wilson lines are equivalent
to products of framed ones), and the resulting commuting projector Hamiltonian
will be local. The Hilbert space at each site would however be infinite-
dimensional, and the resulting theory may end up being gapless — see Refs.
[23, 24] for an in-depth study of similar issues in doubled CS theories. In
addition, one may be able to use our construction to directly compute the
topological entanglement entropy [54, 55, 56] on the lattice, although the
non-local constraints are likely to make this subtle.

One of the most interesting features of CS theory is its robust chiral edge
modes [57, 58]. An obvious next step would be to analyze the edge modes of our
lattice theory on spatial lattice with boundary, and verify that they are
indeed chiral and define a sensible quantum field theory. We hope that this
will give rise to new insights on the problem of discretizing genuinely chiral
bosons and fermions on the lattice (see Refs. [59, 60, 61, 21, 62, 63, 64] for
recent works on the subject). It is also interesting to go up in dimension, as
we did in Ref. [9], and study our CS theory as the boundary of a four-
dimensional symmetry-protected topological (SPT) phase protected by the
$\mathbb{Z}_{k}$ 1-form symmetry. Relatedly, leveraging the results in this
paper one should be able to canonically quantize 4d $U(1)$ lattice gauge
theory with a $\theta$ term to study the Witten effect, electric-magnetic
duality, and more.

Finally, one of our main motivations for studying the odd-level fermionic CS
theories is to establish exact boson-fermion dualities on the lattice. While
in this paper we focused on the pure CS theory, it is straightforward to
introduce charged matter to obtain, for instance, $U(1)_{1}$ coupled to a
scalar. Our analysis already correctly captures the fermionic statistics of
the monopole in that theory (via the spin-charge relation (88)), and
constitutes a significant step towards proving a lattice-level duality with a
Dirac fermion. We leave further exploration of this duality, and others, for
future work.

Acknowledgments

We are grateful to P.S. Hsin for discussions and A. Cherman, T. Dumitrescu, Z.
Komargodski, and S. Seifnashri for comments on the draft. T.J. is supported by
a Julian Schwinger Fellowship from the Mani L. Bhaumik Institute for
Theoretical Physics at UCLA. T.S. is supported by the Royal Society University
Research Fellowship and in part by the STFC consolidated grant ST/T000708/1.

##  Appendix A Quantizing a first-order Lagrangian

Consider the Lagrangian

$L=q_{1}\dot{q}_{2}\;.$ (93)

The canonical momenta are

$\pi_{1}=0\;,\qquad\pi_{2}=q_{1}\;.$ (94)

The equations above are primary, second class constraints, which can be
written in terms of the two functions

$\displaystyle f_{1}(\pi_{1})=\pi_{1}\;,$ (95) $\displaystyle
f_{2}(\pi_{2},q_{1})=\pi_{2}-q_{1}\;.$ (96)

The constrains are indeed second class as the Poisson bracket
$[f_{1},f_{2}]_{PB}=1$. Quantization then proceeds with defining the Dirac
bracket as

$[A,B]_{DB}=[A,B]_{PB}-[A,f_{i}]_{PB}(S^{-1})^{ij}[f_{j},B]_{PB}\;,$ (97)

with $S_{ij}=[f_{i},f_{j}]=-(S^{-1})^{ij}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}$ is an invertable matrix. Since we are imposing the
constraint on $\pi_{1}$ and $\pi_{2}$, the only Dirac bracket we need is

$[q_{2},q_{1}]_{DB}=1\;.$ (98)

We quantize by promoting

$[q_{2},q_{1}]_{DB}\rightarrow[q_{2},q_{1}]=i$ (99)

where $[A,B]$ is the commutator of operators $A$ and $B$. The algebra then
just simply becomes that $q_{2}$ is the conjugate momentum of $q_{1}$.

However now notice that up to a total derivative, we can write the Lagrangian
as

$L=\frac{1}{2}(q_{1}\dot{q}_{2}-q_{2}\dot{q}_{1})\;.$ (100)

The constraints are then given by

$\displaystyle f_{1}=\pi_{1}+\frac{1}{2}q_{2}\;,$ (101) $\displaystyle
f_{2}=\pi_{2}-\frac{1}{2}q_{1}\;.$ (102)

The Poisson bracket between the two constraint again does not vanish

$[f_{1},f_{2}]_{PB}=-\frac{1}{2}[\pi_{1},q_{1}]_{PB}+\frac{1}{2}[q_{2},\pi_{2}]_{PB}=1\;.$
(103)

and so the constraint is second class. The Dirac bracket is defined again in
the same manner and gives the commutation relation

$[q_{2},q_{1}]=i$ (104)

i.e. the same as before.

Let us explicitly compute the Dirac bracket in the first setup for two
functions $A,B$ of $q_{1},q_{2},\pi_{1},\pi_{2}$. Since

$[A,f_{1}]_{PB}=[A,\pi_{1}]_{PB}=\frac{\partial A}{\partial
q_{1}}\;,\qquad[A,f_{2}]_{PB}=\frac{\partial A}{\partial q_{2}}+\frac{\partial
A}{\partial\pi^{1}}\;,$ (105)

we easily get, by direct computation, that the Dirac bracket is given by

$[A,B]_{DB}=\frac{\partial A}{\partial q_{2}}\left(\frac{\partial B}{\partial
q_{1}}+\frac{\partial B}{\partial\pi_{2}}\right)-\frac{\partial B}{\partial
q_{2}}\left(\frac{\partial A}{\partial q_{1}}+\frac{\partial
A}{\partial\pi_{2}}\right)\;.$ (106)

Now, we view $A$ and $B$ as functions of $q_{1,2}$ and $\pi_{1,2}$. The
derivative w.r.t. $\pi_{1}$ does not appear at all, so we can safely set
$\pi_{1}=0$ in the dependence of $A,B$ before or after the Dirac bracket has
been evaluated. However, we must be careful with setting $\pi_{2}=q_{1}$, and
only impose this constraint after the Dirac bracket has been evaluated. In
other words the Dirac bracket is given by

$[A,B]_{DB}=\left[\frac{\partial A}{\partial q_{2}}\left(\frac{\partial
B}{\partial q_{1}}+\frac{\partial B}{\partial\pi_{2}}\right)-\frac{\partial
B}{\partial q_{2}}\left(\frac{\partial A}{\partial q_{1}}+\frac{\partial
A}{\partial\pi_{2}}\right)\right]\Bigg{|}_{\pi_{1}=0,\pi_{2}=q_{1}}\;.$ (107)

Let $F(q_{1},\pi_{2})$ be a function of $q_{1}$ and $\pi_{2}$. The
differential of $F$ is

$dF=\frac{\partial F}{\partial q_{1}}dq_{1}+\frac{\partial
F}{\partial\pi_{2}}d\pi_{2}\;.$ (108)

Now we want to set $q_{1}=\pi_{2}$ and look for a differential of
$F(q_{1},q_{1})$. We find that it is given by

$\frac{dF(q_{1},q_{1})}{dq_{1}}=\frac{\partial F}{\partial
q_{1}}\Big{|}_{\pi_{2}=q_{1}}+\frac{\partial
F}{\partial\pi_{2}}\Big{|}_{\pi_{2}=q_{1}}\;.$ (109)

Hence we get that setting the constraint $\pi_{2}=q_{1}$ before
differentiation (which we denote by $A\big{|}$ and $B\big{|}$) renders the
Dirac bracket between $A$ and $B$ to

$[A,B]_{DB}=\frac{\partial A\big{|}}{\partial q_{2}}\frac{\partial
B\big{|}}{\partial q_{1}}-\frac{\partial B\big{|}}{\partial
q_{2}}\frac{\partial A\big{|}}{\partial q_{1}}\;.$ (110)

In other words, we can really just set $q_{1}$ to be the conjugate momentum of
$q_{2}$.

This discussion can be directly applied to our lattice action (3). The
momentum space action (11) leads to the constraint on canonical momenta

$\Phi_{\mathbf{p},i}=\Pi_{\mathbf{p},i}+\frac{1}{2}K(\mathbf{p})_{ij}\,\tilde{a}_{-\mathbf{p},j}\,.$
(111)

The above constraint will have a non-vanishing Poisson bracket if
$K(\mathbf{p})_{ij}\not=0$, and constitutes a second-class constraint
requiring the Dirac bracket. Proceeding with the modes for which
$\mathbf{p}\not\in\mathcal{P}_{\text{staggered}}$. The Poisson bracket between
the remaining constraints is

$\begin{split}F_{\mathbf{p}\mathbf{p}^{\prime},ii^{\prime}}\equiv[\Phi_{\mathbf{p},i},\Phi_{\mathbf{p}^{\prime},i^{\prime}}]_{\text{PB}}&=\frac{1}{2}K(\mathbf{p})_{ij}\,[\tilde{a}_{-\mathbf{p},j},\Pi_{\mathbf{p}^{\prime},i^{\prime}}]_{\text{PB}}+\frac{1}{2}K(\mathbf{p}^{\prime})_{i^{\prime}j^{\prime}}\,[\Pi_{\mathbf{p},i},\tilde{a}_{-\mathbf{p}^{\prime},j^{\prime}}]_{\text{PB}}\\\
&=\frac{1}{2}\delta_{-\mathbf{p},\mathbf{p}^{\prime}}K(\mathbf{p})_{ii^{\prime}}-\frac{1}{2}\delta_{\mathbf{p},-\mathbf{p}^{\prime}}K(\mathbf{p}^{\prime})_{i^{\prime}i}=\delta_{\mathbf{p},-\mathbf{p}^{\prime}}K(\mathbf{p})_{ii^{\prime}}\,.\end{split}$
(112)

To consistently quantize the system we have to use Dirac brackets. The inverse
of the matrix of constraints (continuing to assume
$\mathbf{p}\not\in\mathcal{P}_{\text{staggered}}$) is

$F^{-1}_{\mathbf{p}\mathbf{p}^{\prime},ii^{\prime}}=-\frac{1}{2}\left(\frac{4\pi}{k}\right)^{2}\frac{1}{1+\cos(p_{1}+p_{2})}\delta_{\mathbf{p},-\mathbf{p}^{\prime}}K(-\mathbf{p})_{ii^{\prime}}\,,$
(113)

so the Dirac bracket between two momentum modes is

$\begin{split}[\tilde{a}_{\mathbf{p},i},\tilde{a}_{\mathbf{q},j}]_{\text{DB}}&=[\tilde{a}_{\mathbf{p},i},\tilde{a}_{\mathbf{q},j}]_{\text{PB}}-[\tilde{a}_{\mathbf{p},i},\Phi_{\mathbf{p}^{\prime},i^{\prime}}]_{\text{PB}}\,F^{-1}_{\mathbf{p}^{\prime}\mathbf{q}^{\prime},i^{\prime}j^{\prime}}\,[\Phi_{\mathbf{q}^{\prime},j^{\prime}},\tilde{a}_{\mathbf{q},j}]_{\text{PB}}\\\
&=0-(\delta_{\mathbf{p},\mathbf{p}^{\prime}}\delta_{ii^{\prime}})\frac{1}{2}\left(\frac{4\pi}{k}\right)^{2}\frac{1}{1+\cos(p_{1}^{\prime}+p_{2}^{\prime})}\delta_{\mathbf{p}^{\prime},-\mathbf{q}^{\prime}}K(-\mathbf{p}^{\prime})_{i^{\prime}j^{\prime}}\delta_{\mathbf{q},\mathbf{q}^{\prime}}\delta_{jj^{\prime}}\\\
&=-\frac{1}{2}\left(\frac{4\pi}{k}\right)^{2}\frac{1}{1+\cos(p_{1}+p_{2})}\delta_{\mathbf{p},-\mathbf{q}}K(-\mathbf{p})_{ij}=\delta_{\mathbf{p},-\mathbf{q}}(K(-\mathbf{p})^{-1})_{ij}\,.\end{split}$
(114)

Upon quantization this reproduces Eq. (17). In the above analysis we did not
include the Gauss law constraint, which despite being second-class, with

$[\tilde{\mathcal{G}}_{\mathbf{p}},\Phi_{\mathbf{q},i}]_{\text{PB}}=(1+e^{i(p_{1}+p_{2})})\delta_{\mathbf{p},\mathbf{q}}\,\epsilon_{ij}(1-e^{ip_{j}})\,,$
(115)

only involves the coordinates and hence does not contribute to the Dirac
bracket.

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