# Gauging $\mathrm{U}(1)$ symmetry in (2+1)d topological phases

Meng Cheng  Department of Physics, Yale University, New Haven, CT 06511-8499,
USA  Chao-Ming Jian  Department of Physics, Cornell University, Ithaca, NY
14853, USA

###### Abstract

We study the gauging of a global U(1) symmetry in a gapped system in (2+1)d.
The gauging procedure has been well-understood for a finite global symmetry
group, which leads to a new gapped phase with emergent gauge structure and can
be described algebraically using the mathematical framework of modular tensor
category (MTC). We develop a categorical description of U(1) gauging in a MTC,
taking into account the dynamics of U(1) gauge field absent in the finite
group case. When the ungauged system has a non-zero Hall conductance, the
gauged theory remains gapped and we determine the complete set of anyon data
for the gauged theory. On the other hand, when the Hall conductance vanishes,
we argue that gauging has the same effect of condensing a special Abelian
anyon nucleated by inserting $2\pi$ U(1) flux. We apply our procedure to the
SU(2)k MTCs and derive the full MTC data for the $\mathbb{Z}_{k}$ parafermion
MTCs. We also discuss a dual U(1) symmetry that emerges after the original
U(1) symmetry of an MTC is gauged.

##  I Introduction

For a quantum many-body system with a global symmetry $G$, coupling to a
background $G$ gauge field is often a particularly effective way to probe the
symmetry property of the ground state. When the gauge field becomes dynamical,
a new quantum phase with $G$ gauge structure emerges. This gauging
construction/approach has played a key role in the recent advances in the
classifications of symmetry-protected topological (SPT) phases and symmetry-
enriched topological (SET) phases Barkeshli  _et al._ (2019); Tarantino  _et
al._ (2016); Teo  _et al._ (2015); Lan  _et al._ (2017); Kong  _et al._
(2020), and more generally has illuminated many new relations between
seemingly different quantum field theories Seiberg  _et al._ (2016).

Roughly speaking, gauging a $G$ symmetry modifies the theory in two ways:
first, new excitations carrying $G$ fluxes are introduced. Second, the Gauss’s
law is imposed and only $G$-invariant states are kept in the Hilbert space.
For a finite $G$, at low energy the $G$ gauge field stays completely flat, and
as a result the local dynamics is essentially unaffected by gauging.
Equivalently, $G$ fluxes in this case are gapped, localized objects. If the
ungauged system is gapped, gauging a finite group symmetry again leads to a
gapped phase.

When the system before gauging is gapped and thus a $G$ symmetry-enriched
topological phase, the gauging procedure can be systematically described
within the mathematical framework of $G$-crossed braided tensor category
Barkeshli  _et al._ (2019). Mathematically, a (2+1)d gapped phase in a bosonic
system is described by a modular tensor category (MTC) $\mathcal{C}$, also
known as the algebraic theory of anyons Kitaev (2006). All universal aspects
of a gapped phase with $G$ symmetry can be captured algebraically by $G$
action on the MTC $\mathcal{C}$ 111Modulo some ambiguity corresponding to
stacking $G$ SPT phases. Then from this information, there is a well-defined
procedure to produce a new MTC representing the gauged system, as described in
Ref. [Barkeshli  _et al._ , 2019].

Gauging a continuous symmetry is however fundamentally different from the
finite group case, due to dynamical considerations. In this note we will be
focusing on U(1) symmetry. It is well-known that in general a compact U(1)
Maxwell gauge theory is completely confined at low energy due to the
proliferation of instantons Polyakov (1977). Here by instanton we mean an
insertion of $2\pi$ magnetic flux, which is a local operator. In other words,
the theory describes a completely trivial phase. This is already dramatically
different from the finite group case, where the “garden variety” gauge theory
is always deconfined, with intrinsic topological order. On the other hand, if
the U(1) gauge theory comes with a Chern-Simons (CS) term, then the theory is
also gapped but with nontrivial topological order.

The purpose of this note is to provide a purely algebraic formulation for
gauging U(1) symmetry in a (2+1)d gapped phase. U(1) symmetry in an MTC
$\mathcal{C}$ can be captured by two pieces of data: an Abelian anyon $v$, the
vison, encoding fractional charges carried by the anyons, and the Hall
conductance $\sigma_{H}$. With this triplet $\mathcal{C},v$ and $\sigma_{H}$,
when $\sigma_{H}\neq 0$ we construct a new MTC $\mathcal{D}$ (i.e. with all
$F$\- and $R$-symbols explicitly determined) which corresponds to the gauged
theory, by formalizing the notion of flux attachment. Mathematically, to
obtain the categorical data of the new MTC $\mathcal{D}$, we first introduce
an infinite category whose anyons are labeled by all combinations of the anyon
labels of the original MTC $\mathcal{C}$ and the compatible U(1) charges.
Fusion rules of anyons in the infinite category are derived from those of the
original MTC $\mathcal{C}$ and the addition of U(1) charges. Braiding-related
data in the infinite category can be obtained from the those of $\mathcal{C}$
plus extra contributions from Aharonov-Bohm phases caused by flux attachment.
The MTC $\mathcal{D}$, namely the gauged theory, can be obtained by condensing
the a transparent anyon, which is directly associated with the vison
$v\in\mathcal{C}$, in this infinite category. We show that the new gauged MTC
$\mathcal{D}$ is also mathematically equivalent to
$\mathcal{C}\boxtimes\mathrm{U}(1)_{-s^{2}\sigma_{H}}\big{|}_{(v,s\sigma_{H})}$
which is derived from an anyon condensation in
$\mathcal{C}\boxtimes\mathrm{U}(1)_{-s^{2}\sigma_{H}}$. Here, $s$ is the
minimal integer such that $s$ copies of the vison $v$ fuse into a trivial
anyon. More detailed explanation of this notation will be given later. When
$\sigma_{H}=0$, we argue that gauging U(1) amounts to condensing the vison
$v$, which can be understood as the consequence of Polyakov’s mechanism for
confinement. We also discuss in this note an emergent U(1)dual symmetry of the
gauged theory $\mathcal{D}$ in both the case with $\sigma_{H}\neq 0$ and the
case with $\sigma_{H}=0$.

##  II Gauging U(1) in an Abelian topological phase

Let us first study gauging of an Abelian topological phase. It is known that
any Abelian topological order in (2+1)d can be represented by a
$\mathrm{U}(1)^{N}$ CS theory Wen and Zee (1992); Cano  _et al._ (2014). When
the (2+1)d Abelian topological order has a global U(1) symmetry, we can couple
the $\mathrm{U}(1)^{N}$ CS theory to the background U(1) gauge field
associated with the global U(1) symmetry:

$\mathcal{L}=-\frac{K_{IJ}}{4\pi}a_{I}da_{J}+\frac{t_{I}}{2\pi}Ada_{I},$ (1)

where $a_{I=1,2,...,N}$ denote the dynamical U(1) gauge fields in the
$\mathrm{U}(1)^{N}$ CS theory, and $A$ is the background U(1) gauge field.
Here, $K$ is an $N\times N$ non-degenerate symmetric integer matrix. In this
paper, we only focus on topological orders in bosonic systems. Hence, the
diagonal entries of $K$ are all required to be even. The coupling between
$a_{I}$ and $A$ is determined by the charge vector
$\mathbf{t}=(t_{1},t_{2},\dots,t_{N})^{\mathsf{T}}$. We will assume that the
charge vector is primitive, i.e. $\gcd(t_{1},t_{2},\dots,t_{N})=1$. Formally
integrating out $a_{I}$’s one readily sees that the system has a Hall
conductance $\sigma_{H}=\mathbf{t}^{\mathsf{T}}K^{-1}\mathbf{t}$.

A quasiparticle or an anyon in the Abelian topological order described by Eq.
(1) can be labeled by a $N$-component integer vector
$\mathbf{l}\in\mathbb{Z}^{N}$ that specifies the gauge charge of this anyon
under the $\mathrm{U}(1)^{N}$ gauge group. Two anyons $\mathbf{l}$ and
$\mathbf{l}^{\prime}$ are topologically equivalent if
$\mathbf{l}-\mathbf{l}^{\prime}=K\mathbf{m}$ for some
$\mathbf{m}\in\mathbb{Z}^{N}$. It implies that quasiparticles that are labeled
by $K\mathbf{m}$ for any $\mathbf{m}\in\mathbb{Z}^{N}$ are trivial, namely
they are local excitations and are topologically equivalent to the vacuum. The
fusion rule of the anyons in theory Eq. (1) is simply given by the addition of
the gauge charges under the $\mathrm{U}(1)^{N}$ gauge group. All anyons in
this theory are Abelian. The topological twist factor of the anyon
$\mathbf{l}$ is given by
$\theta_{\mathbf{l}}=e^{\mathrm{i}\pi\mathbf{l}^{\mathsf{T}}K^{-1}\mathbf{l}}$.
The braiding statistics between the Abelian anyons $\mathbf{l}$ and
$\mathbf{l}^{\prime}$ is given by
$M_{\mathbf{l}\mathbf{l}^{\prime}}=e^{\mathrm{i}2\pi\mathbf{l}^{\mathsf{T}}K^{-1}\mathbf{l}^{\prime}}$.
In addition, one can show that the chiral central charge $c_{-}$ of the system
is given by the signature of the matrix $K$, i.e.
$c_{-}=\operatorname{sig}K=r_{+}-r_{-}$, where $r_{\pm}$ is the number of
positive/negative eigenvalues of $K$.

The coupling to the U(1) background gauge field $A$ in Eq. (1) specifies how
the global U(1) symmetry acts on this Abelian topological order. In
particular, it implies, via the equation of motion, that the anyon
$\mathbf{l}$ carries charge
$Q_{\mathbf{l}}=\mathbf{t}^{\mathsf{T}}K^{-1}\mathbf{l}$ under the global U(1)
symmetry. Also, we observe that the insertion of a $2\pi$ flux of $A$ induces
a specific anyon, the vison, which is given by $v=\mathbf{t}$. Notice that
identity that

$\displaystyle M_{v\mathbf{l}}=e^{\mathrm{i}2\pi Q_{\mathbf{l}}}$ (2)

for any anyon $\mathbf{l}\in\mathbb{Z}^{N}$. Physically, it means that we can
reinterpret the braiding statistics between the vison $v$ and the anyon
$\mathbf{l}$ as the Aharonov-Bohm phase between the $2\pi$-flux of $A$ (that
is the vison) and the U(1) symmetry charge $Q_{\mathbf{l}}$ carried by the
anyon $\mathbf{l}$. Notice that the charge $Q_{\mathbf{l}}$ is fractional if
and only if the braiding statistics $M_{v\mathbf{l}}$ is non-trivial.

When we gauge the U(1) global symmetry in the theory Eq. (1), we promote $A$
to a dynamical U(1) gauge field. The resulting theory is a
$\mathrm{U}(1)^{N+1}$ CS theory with a new $K$-matrix:

$\tilde{K}=\begin{pmatrix}K&-\mathbf{t}\\\
-\mathbf{t}^{\mathsf{T}}&0\end{pmatrix},$ (3)

whose determinant is

$\det\tilde{K}=\mathbf{t}^{\mathsf{T}}K^{-1}\mathbf{t}\det K=\sigma_{H}\det
K.$ (4)

Clearly, the physics of the gauged theory crucially depends on whether
$\sigma_{H}$ vanishes or not. We will discuss the two cases separately.

###  II.1 $\sigma_{H}\neq 0$

Let us first consider $\sigma_{H}\neq 0$ which implies that $\det\tilde{K}\neq
0$. Therefore, the gauged system is a gapped system whose Abelian topological
order is described by the new $K$-matrix $\tilde{K}$ in Eq. (3).

First, we observe that with $\sigma_{H}\neq 0$, the signature of $\tilde{K}$
is

$\operatorname{sig}(\tilde{K})=\operatorname{sig}(K)-\operatorname{sgn}\sigma_{H}.$
(5)

This follows from the following identity:

$W^{\mathsf{T}}\tilde{K}W=\begin{pmatrix}K&0\\\
0&-\sigma_{H}\end{pmatrix},\>W=\begin{pmatrix}\mathbf{1}&K^{-1}\mathbf{t}\\\
0&{1}\end{pmatrix},$ (6)

and the fact that the signature is invariant under invertible similarity
transformation. Since the signature of the $K$-matrix is the chiral central
charge of the topological phase, we find that

$\displaystyle c_{-}^{\prime}=c_{-}-\operatorname{sgn}\sigma_{H}$ (7)

where $c_{-}=\operatorname{sig}({K})$ is the chiral central charge before
gauging the U(1) symmetry and $c_{-}^{\prime}=\operatorname{sig}(\tilde{K})$
is that of the gauged system.

While the gauged theory is basically given by the $\tilde{K}$ matrix in Eq.
(3), in the following we analyze the anyon content of theory, in particular
how they are related to anyons in the theory before gauging, in a manner that
does not rely on the explicit $\mathrm{U}(1)^{N}$ CS field theory
construction. As will be shown later, the result can be easily adopted to more
general topological phases.

Anyons in the gauged theory can be represented by $(N+1)$-component integer
vectors $\begin{pmatrix}\mathbf{l}\\\ q\end{pmatrix}$, where
$\mathbf{l}\in\mathbb{Z}^{N}$ labels the anyons in the original ungauged
theory (i.e. gauge charges under the gauge fields $a_{I=1,2,...,N}$) and
$q\in\mathbb{Z}$ can be understood as an additional charge under $A$ attached
to the anyon $\mathbf{l}$. We can represent all local excitations in the
following form

$\tilde{K}\begin{pmatrix}\mathbf{m}\\\
n\end{pmatrix}=\begin{pmatrix}K\mathbf{m}-n\mathbf{t}\\\
-Q_{K\mathbf{m}}\end{pmatrix}.$ (8)

where $\mathbf{m}\in\mathbb{Z}^{N}$ and $n\in\mathbb{Z}$. After gauging the
U(1) symmetry, there are two kinds of local excitations. The first kind is of
the form $\begin{pmatrix}K\mathbf{m}\\\ -Q_{K\mathbf{m}}\end{pmatrix}$. In
particular, we note that the additional charge under $A$ attached is
$-Q_{K\mathbf{m}}$, exactly canceling the the original $A$-charge
$Q_{K\mathbf{m}}$ carried by $K\mathbf{m}$ to form a charge-neutral object
under $A$. The other type of local excitations is generated by
$\begin{pmatrix}\mathbf{t}\\\ 0\end{pmatrix}$, which is the vison of the
original theory, with no additional $A$-charge attached.

A useful result is the inverse of the new $K$-matrix:

$\tilde{K}^{-1}=\begin{pmatrix}{K^{-1}-\sigma^{-1}_{H}K^{-1}\mathbf{t}\mathbf{t}^{\mathsf{T}}K^{-1}}&-\sigma^{-1}_{H}K^{-1}\mathbf{t}\\\
-\sigma^{-1}_{H}\mathbf{t}^{\mathsf{T}}K^{-1}&-\sigma_{H}^{-1}\end{pmatrix}.$
(9)

Therefore, in the theory after we gauged the U(1) symmetry, the topological
twist factor for the anyon $\mathbf{p}=\begin{pmatrix}\mathbf{l}\\\
q\end{pmatrix}$ is then given by

$\theta_{\mathbf{p}}={\exp\left(\mathrm{i}\pi\mathbf{l}^{\mathsf{T}}K^{-1}\mathbf{l}-\mathrm{i}\pi\sigma_{H}^{-1}(q+Q_{\mathbf{l}})^{2}\right)=\theta_{\mathbf{l}}e^{-\frac{\mathrm{i}\pi}{\sigma_{H}}(q+Q_{\mathbf{l}})^{2}}}$
(10)

Here,
$\theta_{\mathbf{l}}=e^{\mathrm{i}\pi\mathbf{l}^{\mathsf{T}}K^{-1}\mathbf{l}}$
is the topological twist factor of the anyon $\mathbf{l}$ in the ungauged
theory. Also, we recognize $q+Q_{\mathbf{l}}$ as the total charge under $A$
carried by the excitation $\mathbf{p}=\begin{pmatrix}\mathbf{l}\\\
q\end{pmatrix}$, and the additional phase factor
$e^{-\frac{\mathrm{i}\pi}{\sigma_{H}}(q+Q_{\mathbf{l}})^{2}}$ in can be
understood as Aharonov-Bohm-like phase from flux attachment for the gauge
field $A$. The braiding statistics $M_{\mathbf{p}\mathbf{p}^{\prime}}$ of the
gauged theory between the anyons $\mathbf{p}=\begin{pmatrix}\mathbf{l}\\\
q\end{pmatrix}$ and $\mathbf{p}^{\prime}=\begin{pmatrix}\mathbf{l}^{\prime}\\\
q^{\prime}\end{pmatrix}$ has a similar structure:

$M_{\mathbf{p}\mathbf{p}^{\prime}}=e^{\mathrm{i}2\pi\mathbf{p}^{\mathsf{T}}\tilde{K}^{-1}\mathbf{p}^{\prime}}=M_{\mathbf{l}\mathbf{l}^{\prime}}e^{-\frac{\mathrm{i}2\pi}{\sigma_{H}}(q+Q_{\mathbf{l}})(q^{\prime}+Q_{\mathbf{l}}^{\prime})}$
(11)

which is the product of the braiding statistics
$M_{\mathbf{l}\mathbf{l}^{\prime}}$ between anyons $\mathbf{l}$ and
$\mathbf{l}^{\prime}$ in the ungauged theory and an extra phase factor
$e^{-\frac{\mathrm{i}2\pi}{\sigma_{H}}(q+Q_{\mathbf{l}})(q^{\prime}+Q_{\mathbf{l}}^{\prime})}$
due to flux attachment of $A$.

###  II.2 $\sigma_{H}=0$

When the Hall conductance is zero, the new $K$-matrix has a vanishing
determinant, namely $\det\tilde{K}=0$. In this case, the corresponding
$\mathrm{U}(1)^{N+1}$ CS theory by itself is not a valid description of a
topological phase. To see why this is the case, when $\det\tilde{K}=0$, there
exist null vectors $\mathbf{v}$ such that $\tilde{K}\mathbf{v}=\mathbf{0}$. It
is easy to show that the null space (over real numbers) is generated by the
following vector:

$\begin{pmatrix}K^{-1}\mathbf{t}\\\ 1\end{pmatrix}.$ (12)

One can multiply a certain integer to make it integral, and we call the
minimal such integral vector $\mathbf{v}_{0}$. It is then always possible to
find an invertible transformation $W$ such that by a change of variable
$a=Wa^{\prime}$,

$W^{\mathsf{T}}\tilde{K}W=\begin{pmatrix}0&0\\\
0&\tilde{K}^{\prime}\end{pmatrix},$ (13)

where $\tilde{K}^{\prime}$ is non-degenerate. Denote the gauge field
corresponding to the $0$ corner by $a_{1}^{\prime}$, and the corresponding
vector $\mathbf{e}_{1}^{\prime}=(1,0,0,\dots)$. Because there is no Chern-
Simons term for $a_{1}^{\prime}$, on general grounds we should include a
Maxwell term for $a_{1}^{\prime}$. Then due to Polyakov mechanism
$a_{1}^{\prime}$ becomes confined, and because $a_{1}^{\prime}$ has no
(topological) coupling to the remaining gauge fields we can safely ignore
$a_{1}^{\prime}$. The “remaining” non-degenerate $K$-matrix
$\tilde{K}^{\prime}$ should describe the resulting topological order after the
gauging of U(1).

While such invertible transformation $W$ always exists, the explicit
expression and the resulting $\tilde{K}^{\prime}$ are often quite cumbersome
and not particularly enlightening. In the following we use an alternative
formulation of Abelian CS theory to sidestep the need to find
$\tilde{K}^{\prime}$ explicitly. In this formulation, we think of the theory
described by a non-degenerate $K$-matrix $K$ as an integral lattice
$L^{K}=\mathbb{Z}^{\mathrm{dim}K}$ where each vector represents an excitation.
The lattice is equipped with a symmetric bilinear form given by $K^{-1}$. Then
we form the quotient $L^{K}/L^{K}_{\text{loc}}$ where
$L^{K}_{\text{loc}}=\\{K\mathbf{l}|\mathbf{l}\in L^{K}\\}$, which is a finite
Abelian group with a non-degenerate quadratic form. First, we observe that
finding the resulting topological order $\tilde{K}^{\prime}$ is the same as
determining the sub-lattice orthogonal to the vector $\mathbf{v}_{0}$. More
explicitly, a vector $\mathbf{l}$ in the original basis becomes
$W^{-1}\mathbf{l}$ after basis transformation, which is orthogonal to
$\mathbf{e}_{1}^{\prime}=W^{-1}\mathbf{v}_{0}$ if and only if
$\mathbf{l}^{\mathsf{T}}\mathbf{v}_{0}=0$. Therefore, a general vector
$\begin{pmatrix}\mathbf{x}\\\ n\end{pmatrix}$ is orthogonal to
$\mathbf{v}_{0}$ if and only if $\mathbf{x}^{\mathsf{T}}K^{-1}\mathbf{t}=-n$.
In other words, under the gauge field $A$, the charge of the anyon
corresponding to $\mathbf{x}$ in the ungauged theory has to be an integer,
which can be made zero by attaching $-n$ local $A$-charges and then survives
the confinement of $a_{1}^{\prime}$. Physically, this is what one expects from
“condensing” the vison (i.e. $2\pi$ flux) in the original ungauged theory, as
the condensation should confine all anyons which braid non-trivially with
$\mathbf{t}$, i.e. carrying fractional charge under $A$. It is useful to
notice that the orthogonality condition uniquely determines the $(N+1)$-th
component $n$ in terms of the first $N$ components $\mathbf{x}$. In other
words, the sublattice of $L^{\tilde{K}}$ orthogonal to $\mathbf{v}_{0}$ is
actually isomorphic to a subspace of the original lattice $L^{K}$ in which all
vectors have integer charge: $L_{0}^{K}=\\{\mathbf{l}|\mathbf{l}\in
L^{K},Q_{\mathbf{l}}\in\mathbb{Z}\\}$.

Now that we have determined the space orthogonal to $\mathbf{v}_{0}$, it is
still necessary to quotient out the local excitations. It is easy to see that
the two kinds of local excitations, one in the form of
$\begin{pmatrix}K\mathbf{m}\\\ -Q_{K\mathbf{m}}\end{pmatrix}$ and the other
generated by $\begin{pmatrix}\mathbf{t}\\\ 0\end{pmatrix}$, are both in the
orthogonal space $L_{0}^{K}$. In fact, they only differ from
$L_{\text{loc}}^{K}$ by the inclusion of $\mathbf{t}$. So in the end we find
that the topological order after the gauge the U(1) symmetry corresponds to
the quotient

$L_{0}^{K}/(L_{\text{loc}}^{K}\oplus\mathbb{Z}\mathbf{t}),$ (14)

where $\mathbb{Z}\mathbf{t}$ is the sublattice consist of any integer multiple
of $\mathbf{t}$. $L_{\text{loc}}^{K}\oplus\mathbb{Z}\mathbf{t}$ denotes the
lattice generated by all the basis vectors of $L_{\text{loc}}^{K}$ and the
vector $\mathbf{t}$. We readily observe that this is exactly the result of
condensing the vison $\mathbf{t}$ in the original ungauged theory:
$L_{0}^{K}/L_{\text{loc}}^{K}$ corresponds to all Abelian anyon types which
braid trivially with $\mathbf{t}$. Further modding out $\mathbb{Z}\mathbf{t}$
identifies different anyon types that are related to each other by fusing with
(multiples of) $\mathbf{t}$.

##  III Gauging U(1) symmetry in a general MTC

We now proceed to describe U(1) gauging in a general gapped phase in (2+1)d.
It is widely believed that bosonic gapped phases without any global symmetry
in (2+1)d are completely classified by $(\mathcal{C},c_{-})$, where
$\mathcal{C}$ is an MTC which encodes all universal properties of anyonic
quasiparticles in the bulk, and $c_{-}$ is the chiral central charge of the
edge. For a review of MTC in the context of (2+1)d topological phase, we refer
the readers to Ref. [Kitaev, 2006]. In the following, we will use the terms
“quasiparticles” and “anyons” interchangeably. We will denote anyon types by
$a,b,c,\dots$, and the trivial anyon type (i.e. all local excitations) by $1$.

To perform gauging, we need to first review how the global U(1) symmetry acts
on the topological phase Cheng  _et al._ (2016). We will follow the general
theoretical framework established in Ref. [Barkeshli  _et al._ , 2019].
Without loss of generality, we can assume that the fundamental physical charge
of the U(1) symmetry is $1$. In a U(1)-symmetric topological phase, each
quasiparticle $x$ carries a U(1) charge $Q_{x}$. Note that only $Q_{x}\text{
mod }1$ is determined by the topological anyon type of $x$, since we may
attach local U(1)-charged excitations to change $Q_{x}$ by any integer. By
definition, the trivial anyon type $1$, representing all local bosonic
excitations, have $Q_{1}=0$. Here, notice the fact that U(1) being a
continuous connected group cannot permute the anyon labels which are
intrinsically discrete. The U(1) symmetry action on the topological phase is
fully characterized by the fractional charge of each anyon. The charges
$Q_{a}$ of the anyons must satisfy

$Q_{a}+Q_{b}\equiv Q_{c}\text{ mod }1,\text{ if }N_{ab}^{c}>0.$ (15)

Here $N_{ab}^{c}$ is the fusion coefficient, i.e. the multiplicity of anyon
type $c$ in the fusion $a\times b$. As shown in Ref. [Barkeshli  _et al._ ,
2019], there exists a unique Abelian anyon $v$ such that

$\displaystyle e^{2\pi\mathrm{i}Q_{a}}=M_{av}$ (16)

for all anyon types $a$, where $M_{av}$ is the braiding phase between $a$ and
$v$. Physically, $v$ is the excitation created by $2\pi$ flux insertion in
this topological phase. By a straightforward generalization of the celebrated
Laughlin argument, the charge $Q_{v}$ is given by $Q_{v}=\sigma_{H}$ mod $1$,
where $\sigma_{H}$ is the dimensionless Hall conductance. Formally, the Hall
response is captured by the effective action

$S=\frac{\sigma_{H}}{4\pi}\int AdA,$ (17)

where $A$ is the background U(1) gauge field. In addition, one can show that
$\sigma_{H}$ and $v$ are related by $e^{\mathrm{i}\pi\sigma_{H}}=\theta_{v}$
Kapustin and Sopenko (2020); Senthil and Levin (2013), so $v$ determines
$\sigma_{H}$ up to an even integer. The ambiguity is exactly the Hall
conductance of bosonic integer quantum Hall states Chen  _et al._ (2013); Lu
and Vishwanath (2012); Senthil and Levin (2013). To summarize, the U(1)
symmetry enrichment is fully determined by the vison $v$ and the Hall
conductance $\sigma_{H}$.

We define $s$ to be the minimal positive integer such that $v^{s}=1$, namely a
minimal of $s$ copies of the anyon $v$ can fuse into the trivial anyon $1$. It
immediately follows that all fractional charges of the anyons are integer
multiples of $1/s$. We further prove that there must exist a anyon carrying
charge $1/s$, i.e. the minimal charge among the anyons is $e^{*}=1/s$. Since
$v^{s}$ is trivial, it also implies that $s^{2}\sigma_{H}$ must be an even
integer because the topological twist factor $\theta_{v^{s}}=e^{\mathrm{i}\pi
s^{2}\sigma_{H}}$ must be 1 for a trivial anyon. Also, note that
$s\sigma_{H}$, being the U(1) charge carried by $v^{s}$, must be an integer.

Just like in the Abelian case, we need to treat $\sigma_{H}\neq 0$ and
$\sigma_{H}=0$ separately. First we assume $\sigma_{H}\neq 0$. Gauging the
U(1) symmetry means that the background U(1) gauge field $A$ is promoted to a
dynamical field. The Hall conductance implies that flux of $A$ must be
attached to $A$ charge. In other words, the gauged theory only allow states
that satisfy the constraint $\sigma_{H}\Phi+Q=0$, which can be obtained from
the equation of motion of Eq. (17). Here, $\Phi$ is the flux of the gauge
field $A$. Thus a quasiparticle with charge $Q$ is attached a flux
$-\frac{2\pi Q}{\sigma_{H}}$. This flux attachment changes the topological
twist factor by a factor of
$\exp\left(-\mathrm{i}\pi\frac{Q^{2}}{\sigma_{H}}\right)$ via the Aharonov-
Bohm effect between the flux and the charge of $A$. Based on these
observations, we now describe the anyon content of the gauged system.

First of all, due to flux attachment it is sufficient to keep track of the
charge of an excitation. In the ungauged theory, the anyon type
$a\in\mathcal{C}$ only determines the charge $Q_{a}$ mod 1, and excitations
belonging to the same anyon type $a$ can have any value of charge $Q_{a}+n$
where $n\in\mathbb{Z}$. After gauging, naively all these different charges
become distinct superselection sectors. Therefore, it will prove to be
convenient to first enlarge our set of anyons by including explicitly the
charge quantum number. Namely, we now consider all possible excitations
$(a,Q_{a})$, subject to the constraint $e^{2\pi\mathrm{i}Q_{a}}=M_{av}$.
Formally we are working with a much larger theory with (countably) infinite
many types of particles 222Technically speaking, this enlarged category should
satisfy all axioms of ribbon fusion category except the finiteness condition.
We will denote this intermediate category by $\mathcal{C}^{\prime}$.

One reason to take this detour is that the topological data of this enlarged
category $\mathcal{C}^{\prime}$ can be explicitly written down, which we
present now. First of all, the identity object is $(1,0)$, so the anti-
particle of $(a,Q_{a})$ is $(\bar{a},-Q_{a})$, where $\bar{a}$ denotes the
anti-particle of the anyon $a$ within the MTC $\mathcal{C}$. The fusion rules
are given by

$(a,Q_{a})\times(b,Q_{b})=\sum_{c}N_{ab}^{c}(c,Q_{a}+Q_{b}).$ (18)

The flux attachment also modifies braiding and exchange statistics between
anyons, by additional phase factors from Aharonov-Bohm effect between charge
and flux. They can be computed solely from the effective CS response. The
$S$-matrix of the intermediate category $\mathcal{C}^{\prime}$ is hence given
by

$S_{(a,Q_{a}),(b,Q_{b})}=S_{ab}e^{2\pi\mathrm{i}\frac{Q_{a}Q_{b}}{\sigma_{H}}}.$
(19)

with $S_{ab}$ the topological $S$-matrix of the ungauged theory $\mathcal{C}$.
The topological twist factors in the category $\mathcal{C}^{\prime}$ become

$\theta_{(a,Q_{a})}=\theta_{a}e^{-\pi\mathrm{i}\frac{Q_{a}^{2}}{\sigma_{H}}},$
(20)

where $\theta_{a}$ is the topological twist factor of the anyon $a$ in the
ungauged theory. It is straightforward to check that the $S$-matrix,
topological twist factors and fusion rules satisfy compatibility conditions
expected from axioms of braided fusion categories, even though the number of
anyon types is now infinity.

We argue that the flux attachment induced by the dynamics of $A$ does not lead
to non-trivial contributions to the $F$-symbols of $\mathcal{C}^{\prime}$,
since the exact charges are fused. Therefore, we expect the following $F$\-
and $R$-symbols for this enlarged infinite category $\mathcal{C}^{\prime}$:

$\begin{gathered}{}[F^{(a,Q_{a}),(b,Q_{b}),(c,Q_{c})}_{(d,Q_{d})}]_{(e,Q_{e}),(f,Q_{f})}=[F^{abc}_{d}]_{ef}\\\
R^{(a,Q_{a}),(b,Q_{b})}_{(c,Q_{c})}=R^{ab}_{c}e^{-\frac{\pi\mathrm{i}}{\sigma_{H}}Q_{a}Q_{b}}\end{gathered}$
(21)

One can readily show that pentagon and hexagon equations are satisfied, and
the data correctly reproduce the topological $S$-matrix given in Eq. (19) and
the topological twist factors given in Eq. (20).

Next, we truncate the infinite category $\mathcal{C}^{\prime}$ to obtain a
more physical description of the gauged theory. We expect that after gauging,
the system remains gapped (because of a non-zero Hall conductance), so should
be described by a new MTC $\mathcal{D}$. This gauged theory $\mathcal{D}$ will
have different $F$\- and $R$-symbols compared to those of
$\mathcal{C}^{\prime}$ given in Eq. (21) because of the truncation. We begin
with a formal approach: first identify the transparent anyons of the enlarged
infinite category $\mathcal{C}^{\prime}$. Here, transparent anyons are those
anyons that have trivial braiding with every other anyon, so they should be
understood as local excitations (of the gauged system). Suppose $(a,Q_{a})$
braids trivially with all anyons in $\mathcal{C}^{\prime}$. The anyon $a$
should be Abelian. Then

$M_{(a,Q),(b,Q_{b})}=M_{ab}e^{-2\pi\mathrm{i}\frac{Q_{a}Q_{b}}{\sigma_{H}}}=1.$
(22)

So we must have $M_{ab}=e^{2\pi\mathrm{i}\frac{Q_{a}}{\sigma_{H}}Q_{b}}$ for
all $b$ and $Q_{b}$, and apparently the relation is invariant under
$Q_{b}\rightarrow Q_{b}+1$. Thus we find
$e^{2\pi\mathrm{i}\frac{Q_{a}}{\sigma_{H}}}=1$, or $Q_{a}=n\sigma_{H}$.
Because $M_{ab}=e^{2\pi\mathrm{i}nQ_{b}}=M_{v^{n},b}$ for all $b$, by
modularity of $\mathcal{C}$ we must have $a=v^{n}$. Thus the group of
transparent anyons in $\mathcal{C}^{\prime}$ is generated via fusion by the
anyon $(v,\sigma_{H})$ whose self-statistics is bosonic, namely
$\theta_{(v,\sigma_{H})}=1$. Denote $T_{k}=(v^{k},k\sigma_{H})$ for
$k\in\mathbb{Z}$ and note that $T_{k}=T_{k^{\prime}}$ if and only if
$k=k^{\prime}$ (because of a non-zero Hall conductance $\sigma_{H}$). We thus
condense the group of transparent anyons, generated by $T_{1}$, and obtain a
new MTC $\mathcal{D}$, which describes the topological order resulting from
gauging the U(1) symmetry of the original theory $\mathcal{C}$. It is easy to
see that after the condensation the theory contains finitely many anyons.
Observe that $T_{s}=(1,s\sigma_{H})$ is condensed. Therefore the charges in
the category can be restricted to the range $0\leq Q_{a}<s\sigma_{H}$. Thus
for each $a$, there are only a finite number of choices for $Q_{a}$ that need
to be considered. In fact, with proper conventions, one can even restrict
$Q_{a}$ to range $0\leq Q_{a}<\sigma_{H}$ in the gauged theory $\mathcal{D}$
as we can see in the examples later.

Physically, it is quite evident that the transparent particle $(v,\sigma_{H})$
should be interpreted as the vison $v$ but with $-2\pi$ flux of $A$ attached.
Since $v$ is created by the $2\pi$ flux insertion, the composite
$(v,\sigma_{H})$ becomes a local excitation and therefore must be condensed.

Mathematically, the advantage of going through the intermediate steps (i.e.
enlarging the category and then condensing transparent anyons) is that the
approach allows us to write down the full topological data, especially the
$F$\- and $R$-symbols, of the resulting MTC $\mathcal{D}$ explicitly, in terms
of those of the original MTC, and the U(1) symmetry enrichment data $v$ and
$\sigma_{H}$. The details of the procedure are given in App. B.

While the general expressions of the $F$\- and $R$-symbols of the MTC
$\mathcal{D}$ that describes the gauged theory are rather complicated, the
topological $S$-matrix and the topological twist factors of $\mathcal{D}$ can
be simply written down in terms of those of the original MTC $\mathcal{C}$. As
discussed above, the MTC $\mathcal{D}$ can be obtained from condensing
$(v,\sigma_{H})$ in the infinite category $\mathcal{C}^{\prime}$. When
$(v,\sigma_{H})$ is condensed, the anyons $(a,Q_{a})\in\mathcal{C}^{\prime}$
that are only different from each other by the fusion with an integer copies
of $(v,\sigma_{H})$ are identified as a single type of anyon in $\mathcal{D}$.
In other words, the anyons in $\mathcal{C}^{\prime}$ form orbits under the
fusion with $(v^{k},k\sigma_{H})$ for $k\in\mathbb{Z}$. Each different orbit
corresponds to a different type of anyon in $\mathcal{D}$. The topological
twist factor of an anyon in $\mathcal{D}$ is the same as that of any
representative anyon within the corresponding orbit in $\mathcal{C}^{\prime}$,
which can be calculated using Eq. (20). Since one can easily show that an
orbit of anyons in $\mathcal{C}^{\prime}$ shares the same topological twist
factor $\theta_{(a,Q_{a})}$, the choice of representative does not affect the
result. The topological $S$-matrix of the MTC $\mathcal{D}$ can be obtained in
a similar fashion. The $S$-matrix element between two anyons of $\mathcal{D}$
is the same as the $S$-matrix element between their respective representatives
in $\mathcal{C}^{\prime}$ which is given in Eq. (19). An immediate consequence
is that an anyon in $\mathcal{D}$ has the same quantum dimension as its
representative in $\mathcal{C}^{\prime}.$

So far we have treated the category $\mathcal{C}^{\prime}$ as a pure
mathematical device. One may also wonder whether $\mathcal{C}^{\prime}$ has
any physical meaning. In App. F, we provide a “holographic” viewpoint on
gauging, where $\mathcal{C}^{\prime}$ appears as a particular boundary theory
for a (3+1)d Maxwell U(1) gauge theory. Going from $\mathcal{C}^{\prime}$ to
$\mathcal{D}$ then corresponds to the confinement transition in the (3+1)d
bulk.

At this point, the only missing information about the gauged system is the
chiral central charge $c_{-}^{\prime}$. Since we have determined the bulk
anyon data of the gauged system, we can evaluate $c_{-}^{\prime}$ mod 8, by
using the generalized Gauss-Milgram sum. As shown in App. D, we find

$c_{-}^{\prime}=c_{-}-\operatorname{sgn}\sigma_{H}~{}~{}\rm{mod}~{}8.$ (23)

In fact, we believe that $c_{-}^{\prime}=c_{-}-\operatorname{sgn}\sigma_{H}$
holds exactly. In the Abelian case, the relation was proven by the field
theory construction. Now that we’ve obtained all the data of the gauged
theory, our gauging procedure is completed. Note that gauging a finite group
leaves the chiral central charge unchanged Barkeshli  _et al._ (2019), which
is fundamentally different from the case when a U(1) symmetry is gauged.

Moreover, it is interesting to observe that the gauged theory $\mathcal{D}$ is
mathematically equivalent to the following MTC

$\mathcal{C}\boxtimes\mathrm{U}(1)_{-s^{2}\sigma_{H}}\big{|}_{(v,s\sigma_{H})},$
(24)

where we stack the MTC $\mathcal{C}$ with an additional layer of MTC described
by the $\mathrm{U}(1)_{-s^{2}\sigma_{H}}$ CS theory, and condense the
composite anyon
$(v,s\sigma_{H})\in\mathcal{C}\boxtimes\mathrm{U}(1)_{-s^{2}\sigma_{H}}$ with
$v\in\mathcal{C}$ and $s\sigma_{H}\in\mathrm{U}(1)_{-s^{2}\sigma_{H}}$. Such a
MTC
$\mathcal{C}\boxtimes\mathrm{U}(1)_{-s^{2}\sigma_{H}}\big{|}_{(v,s\sigma_{H})}$
can also be viewed as a “hierarchy construction” proposed in Ref. [Lan and
Wen, 2017] which is a categorical formulation of the hierarchy construction of
the fractional quantum Hall states Haldane (1983); Halperin (1984); Bonderson
and Slingerland (2008). In the hierarchy construction, the additional layer
$\mathrm{U}(1)_{-s^{2}\sigma_{H}}$ is also a U(1) SET, with $s\sigma_{H}$
being the vison. Then after condensing the bound state of visons from both
layers, the remaining anyons must be the composite of an anyon
$a\in\mathcal{C}$ and an anyon with a gauge charge $q$ in the U(1) CS theory
satisfying the condition $M_{av}=e^{2\pi\mathrm{i}\frac{q}{s}}$. Note that in
the Abelian layer the anyon $q$ carries U(1) charge $Q_{q}=-q/s$, so by
pairing up $a$ in $\mathcal{C}$ with $q$ in
$\mathrm{U}(1)_{-s^{2}\sigma_{H}}$, after condensation all remaining anyons
are charge-neutral. Another useful observation is that the subcategory of
$\mathcal{C}\boxtimes\mathrm{U}(1)_{-s^{2}\sigma_{H}}$ that braids trivially
with $(v,s\sigma_{H})$ can already be identified with the premodular category
$\mathcal{D}_{\rm int}$ which is obtained in App. B.2 as an intermediate step
towards the full U(1) gauging of $\mathcal{C}$. With this observation, it is
straightforward to check this hierarchy construction indeed gives the MTC
$\mathcal{D}$ which is the outcome of gauging the U(1) symmetry of
$\mathcal{C}$. This result Eq. 24 can be also understood using a gauging
procedure based symmetry group extensionsCheng  _et al._ (2022a).

Below we provide an example of gauging a U(1) symmetry in SU(2)k MTC, and as a
result obtaining all categorical data for $\mathbb{Z}_{k}$ parafermion MTCs.
In the next section, we will discuss the gauging of the U(1) symmetry when the
Hall conductance vanishes, i.e. $\sigma_{H}=0$.

###  III.1 Example: Gauging U(1) SPT phase

Consider the simplest case $\mathcal{C}=\mathrm{Vec}$, i.e. a U(1) bosonic SPT
phase, with Hall conductance $\sigma_{H}=n\in 2\mathbb{Z}$. The only “anyon”
in the MTC $\mathcal{C}=\mathrm{Vec}$ is the trivial one, denoted as “1” in
the following. In this case, the vison $v$ has to be the trivial anyon, namely
$v=1$. Let us assume a positive Hall conductance, i.e. $\sigma_{H}=n>0$ in the
following. The intermediate infinite category $\mathcal{C}^{\prime}$ consists
of all charges $\\{(1,q)\\}$ for $q\in\mathbb{Z}$ (where the first entry “$1$”
denote the trivial anyon in $\mathcal{C}=\mathrm{Vec}$). Condensing
$(v,\sigma_{H})=(1,n)$ in $\mathcal{C}^{\prime}$, we are left with $n$
different anyons $(1,a)$ for $0\leq a<n$ and $a\in\mathbb{Z}$ whose the fusion
rule is given by

$(1,a)\times(1,b)=(1,\lceil a+b\rfloor_{n}),$ (25)

where $\lceil a+b\rfloor_{n}$ is the residue of $a+b$ modulo $n$. Using Eq.
(36) we find that the $F$-symbol of the gauged theory $\mathcal{D}$ is given
by

$F^{abc}=e^{-\frac{\pi\mathrm{i}}{n}a(b+c-\lceil b+c\rfloor_{n})},$ (26)

and $R$\- symbol is given by

$R^{ab}=e^{-\frac{\pi\mathrm{i}}{n}ab}.$ (27)

The resulting MTC $\mathcal{D}$ describes an Abelian topological order. In the
$F$\- and $R$-symbols given above, the suppressed indices are completely
determined via the fusion rule by the indices that are explicitly written. The
resulting MTC data agrees with what is known as the U(1)-n, or
$\mathbb{Z}_{n}^{(-1/2)}$ MTC. One can easily confirm this result using a
field-theoretic approach. Before gauging the U(1) symmetry, the U(1) bosonic
SPT phase simply produces a non-trivial response theory with an action
$S=\frac{n}{4\pi}\int AdA$ for the background $A$ gauge field. The constituent
matter fields of the U(1) bosonic SPT phase have already been safely
integrated out here because the U(1) bosonic SPT state only carries invertible
topological order. When the U(1) symmetry is gauged, $A$ becomes a dynamical
U(1) gauge field. The same action $\frac{n}{4\pi}\int AdA$ now describes the
(2+1)d topological order of U(1)-n.

###  III.2 Example: $\mathbb{Z}_{k}$ parafermion

Consider a (2+1)d topological phase described by the SU(2)k MTC, namely
$\mathcal{C}={\rm SU}(2)_{k}$. There are $k+1$ different types of anyons,
labeled by $j=0,1/2,\cdots,k/2$. Note that there is a single nontrivial
Abelian anyon $k/2$.

This system naturally admits a SO(3) global symmetry: Anyons of type $j$ carry
spin-$j$ representation of SO(3). We now consider gauging a U(1) subgroup of
SO(3), with the vison associated with the 2$\pi$ flux given by $v=k/2$. Since
$\theta_{k/2}=e^{\pi ik/2}$, the Hall conductance is $\sigma_{H}=\frac{k}{2}$
mod $2\mathbb{Z}$. While the formalism is general, the even $k$ case is of
most interest. When $k$ is odd, the anyon $k/2$ is a semion and the whole MTC
factorizes into a “SO(3)k” MTC and the semion theory. The U(1) symmetry only
acts non-trivially on the semion part. So in the following we assume $k$ is
even.

In App. B.3, we carry out the gauging procedure in full detail and derive the
complete set of topological data for the gauged theory $\mathcal{D}$. When
$\sigma_{H}=\frac{k}{2}$, we show that the resulting MTC the $\mathbb{Z}_{k}$
parafermion MTC. The parafermion MTC, initially derived from the
$\mathbb{Z}_{k}$ parafermion conformal field theory introduced by Ref. [Fateev
and Zamolodchikov, 1985], is closely related to the topological order of the
Read-Rezayi fractional quantum Hall states Read and Rezayi (1999). In the
following, we will study how the $\mathbb{Z}_{k}$ parafermion MTC arises from
U(1) gauging in SU(2)k using a field-theoretic formalism, to corroborate the
algebraic approach.

Before gauging, we can describe the U(1) symmetric SU(2)k MTC using a
dynamical SU(2)k Chern-Simons theory. We denote the gauge connection of the
SU(2) gauge field as
$a=\frac{1}{2}\sigma^{1}a^{1}+\frac{1}{2}\sigma^{2}a^{2}+\frac{1}{2}\sigma^{3}a^{3}$
where the Pauli matrices $\frac{1}{2}\sigma^{1,2,3}$ are the generators of the
SU(2) gauge group and the 1-forms $a^{1,2,3}$ are the three associated
components of the SU(2) gauge field. Let the 1-form $A$ denote the background
gauge field associated with the U(1) symmetry.

The anyon $j$ in the SU(2)k MTC corresponds to the spin-$j$ representation of
the SU(2) gauge group. A natural way to assign $j/2$ charge to the anyon $j$
for all $j=0,1,2,...,k/2$ is to consider embedding the U(1) and SU(2) gauge
connections together into a single U(2) gauge field $b$:

$\displaystyle
b=\frac{1}{2}\mathds{1}A+\frac{1}{2}\sigma^{1}a^{1}+\frac{1}{2}\sigma^{2}a^{2}+\frac{1}{2}\sigma^{3}a^{3},$
(28)

where $\mathds{1}$ is the $2\times 2$ identity matrix. The $j=1/2$ anyon which
carries the two-dimensional representation under the U(2) gauge group
naturally carries U(1) charge $1/2$. More generally, an anyon $j$ should carry
a representation under the gauge group U(2) that is given by the symmetric
combination of $2j$ copies of the representation carried by the anyon labeled
by $j=1/2$. Therefore, the anyon $j$ carries U(1) charge $j/2$. Now, we
consider the following Lagrangian that decribes a SU(2)k topological order
under a background U(1) gauge field:

$\displaystyle\mathcal{L}=-\frac{k}{4\pi}\operatorname{Tr}\left(bdb-\frac{2\mathrm{i}}{3}b^{3}\right)+\frac{k}{4\pi}(\operatorname{Tr}b)d(\operatorname{Tr}b),$
(29)

which is also known as the U(2)k,-k Chern-Simons term Seiberg and Witten
(2016); Hsin and Seiberg (2016). Our convention for the Lagrangian of the
U(2)k,-k Chern-Simons term contains an extra overall minus sign compared to
Ref. [Hsin and Seiberg, 2016]. This extra minus sign is due to our convention
of the sign of the chirality of the corresponding MTC with respect to the sign
of the Hall conductance. When we turn off the background U(1) gauge field by
setting $A=0$, the Lagrangian $\mathcal{L}$ above is reduced to
$-\frac{k}{4\pi}\operatorname{Tr}\left(ada-\frac{2\mathrm{i}}{3}a^{3}\right)$
which describes the SU$(2)_{k}$ topological order. With a finite background
U(1) gauge field $A$, this Lagrangian $\mathcal{L}$ effectively contains an
extra term “$\frac{k}{8\pi}AdA$” almost decoupled from the SU$(2)_{k}$ part,
which indicates a Hall conductance of $\sigma_{H}=k/2$. Here, strictly
speaking, “$\frac{k}{8\pi}AdA$” is not truly well-defined on its own and the
U(1) background gauge field can not be fully decoupled from the dynamical
SU(2) gauge field $a$: the embedding of U(1) and SU(2) gauge fields into a
single U(2) gauge field requires an identification between the
$\mathbb{Z}_{2}$ subgroup of U(1) with the center $\mathbb{Z}_{2}$ of SU(2).
Upon gauging the U(1) symmetry of the SU(2)k topological order, the theory
simply becomes a fully dynamical U(2)k,-k Chern-Simons gauge theory. Following
Ref. [Hsin and Seiberg, 2016], the TQFT can be identified as

$\mathrm{U}(2)_{k,-k}\equiv\frac{{\rm SU(2)}_{k}\times{\rm
U}(1)_{-2k}}{\mathbb{Z}_{2}},$ (30)

where the quotient $\mathbb{Z}_{2}$ means that the anyon $(k/2,k)\in{\rm
SU(2)}_{k}\times{\rm U}(1)_{-2k}$ in ${\rm SU(2)}_{k}\times{\rm U}(1)_{-2k}$.
(The notation for the anyons in ${\rm SU(2)}_{k}\times{\rm U}(1)_{-2k}$ will
be explained in detail in the following). We recognize that the right-hand
side of Eq. (30) is exactly the hierarchy construction for $\mathcal{D}$ in
Eq. (24). Below we will explicitly check the equivalence between the hierarchy
construction and the gauging procedure as described in Sec. III (and App. B).

Let’s first unpack the algebraic definition of U(2)k,-k. One can start with
the MTC given by $\mathcal{B}_{0}={\rm SU(2)}_{k}\times{\rm U}(1)_{-2k}$ whose
anyons are labeled by the pair $(j,n)$. Here, $j=0,1,...,k/2$ labels the anyon
type within the SU(2)k sector and $n=0,1,...,2k$ labels the anyon type within
the ${\rm U}(1)_{-2k}$ sector. The fusion rule of the Abelian topological
order ${\rm U}(1)_{-2k}$ is given by $n_{1}\times n_{2}=\lceil
n_{1}+n_{2}\rfloor_{2k}$, where $\lceil\cdot\rfloor_{2k}$ denotes the residue
of $\cdot$ modulo $2k$.

To obtain the $\mathbb{Z}_{k}$ parafermion MTC, one needs to condense the
anyon $(j=k/2,n=k)$ in ${\rm SU(2)}_{k}\times{\rm U}(1)_{-2k}$. Note that the
anyon $(k/2,k)$ has a bosonic self-statistics but is not a transparent anyon
in $\mathcal{B}_{0}$. One can define a sub-category
$\mathcal{B}_{1}\subset\mathcal{B}_{0}$ consists of only anyons in
$\mathcal{B}_{0}$ that braid trivially with the anyon $(k/2,k)$. The anyons in
the sub-category $\mathcal{B}_{1}$ are the anyons $(j,n)$ such that $2j+n\in
2\mathbb{Z}$. The $F$\- and $R$-symbols of category $\mathcal{B}_{1}$ are
given by

$\displaystyle\left(F^{(j_{1},n_{1}),(j_{2},n_{2}),(j_{3},n_{3})}_{(j,n)}\right)_{(j_{12},n_{12}),(j_{23},n_{23})}$
$\displaystyle=\left(F^{j_{1},j_{2},j_{3}}_{j}\right)_{j_{12},j_{23}}e^{-\mathrm{i}\frac{\pi}{2k}n_{1}(n_{2}+n_{3}-\lceil
n_{2}+n_{3}\rfloor_{2k})}.$ $\displaystyle
R^{(j_{1},n_{1}),(j_{2},n_{2})}_{(j,n=\lceil n_{1}+n_{2}\rfloor_{2k})}$
$\displaystyle=R^{j_{1},j_{2}}_{j}e^{-\mathrm{i}\frac{2\pi}{4k}n_{1}n_{2}}$
(31)

Each of the $F$\- and $R$-symbols of the category $\mathcal{B}_{1}$ in Eq.
(31) is a product of two factors.
$\left(F^{j_{1},j_{2},j_{3}}_{j}\right)_{j_{12},j_{23}}$ and
$R^{j_{1},j_{2}}_{j}$ are the $F$\- and $R$-symbols of SU(2)k while the
factors $e^{-\mathrm{i}\frac{\pi}{2k}n_{1}(n_{2}+n_{3}-\lceil
n_{2}+n_{3}\rfloor_{2k})}$ and $e^{-\mathrm{i}\frac{2\pi}{4k}n_{1}n_{2}}$ are
the contributions from the U(1)-2k sector. By definition, the category
$\mathcal{B}_{1}$ is a premodular category with the anyon $(k/2,k)$ the
transparent Abelian anyon. By condensing $(k/2,k)$ in $\mathcal{B}_{1}$, we
can obtain the $\mathbb{Z}_{k}$ parafermion MTC.

Interestingly, before we condense the anyon $(k/2,k)\in\mathcal{B}_{1}$, the
premodular category $\mathcal{B}_{1}$ can already be identified as a category
$\mathcal{D}_{\rm int}$ obtained as an intermediate step of gauging the U(1)
symmetry of SU(2)k. The procedure of gauging the U(1) symmetry of SU(2)k
starts with considering the category $\mathcal{C}^{\prime}$ with infinitely
many anyons $(j,Q)$ defined by the data in Eq. (18) and in Eq. (21). Gauging
the U(1) symmetry amounts to condensing the anyon
$(v,\sigma_{H}=k/2)=(j=k/2,Q=k/2)\in\mathcal{C}^{\prime}$. As explained in
App. B.2, we can first condense the anyon $(1,s\sigma_{H})$ in
$\mathcal{C}^{\prime}$ to obtain an intermediate finite category, denoted as
$\mathcal{D}_{\rm int}$, with $s=2$ due to the $\mathbb{Z}_{2}$ fusion rule of
the anyon $j=k/2$ in SU(2)k. The $F$\- and $R$-symbols of the category
$\mathcal{D}_{\rm int}$ can be calculated using Eq. (40) and Eq. (41). One can
see that the so-obtained $F$\- and $R$-symbols of the category
$\mathcal{D}_{\rm int}$ are the same as the those of the category
$\mathcal{B}_{1}$ shown in Eq. (31) upon identifying the anyon $(j,Q)$ of
$\mathcal{D}_{\rm int}$ as the anyon $(j,n=2Q)$ of $\mathcal{B}_{1}$.
Therefore, the categories $\mathcal{D}_{\rm int}$ and $\mathcal{B}_{1}$ are
completely identical. By further condensing the transparent anyon
$(v,\sigma_{H}=k/2)=(j=k/2,Q=k/2)$ in $\mathcal{D}_{\rm int}$ (or equivalently
the transparent anyon $(j=k/2,k)$ in $\mathcal{B}_{1}$), we can complete the
procedures for gauging the U(1) symmetry of the SU(2)k MTC and, as the result,
obtain the $\mathbb{Z}_{k}$ parafermion MTC.

The simplest example is the case of $k=2$, namely the SU(2)2 MTC. When the
Hall conductance is $\sigma_{H}=k/2=1$, gauging the U(1) symmetry results in
the $\mathbb{Z}_{2}$ parafermion MTC which is more commonly referred to as the
Ising MTC. Interestingly, when we change Hall conductance to $\sigma_{H}=-1$,
the resulting MTC becomes Spin$(5)_{1}$ instead, which is a close relative to
the Ising MTC.

It is not difficult to generalize the discussion to gauging a U(1) symmetry in
SU$(N)_{k}$, which results in a U$(N)_{k,k+Nk^{\prime}}$ MTC (when
$k+Nk^{\prime}\neq 0$ and $k+k^{\prime}$ even). More details can be found in
App. E.

##  IV Gauging $\rm{U}(1)$ symmetry when $\sigma_{H}=0$

We now discuss what happens when $\sigma_{H}$ vanishes. In this case, the
vison $v$ must have a bosonic self-statistics, i.e.
$\theta_{v}=e^{\mathrm{i}\pi\sigma_{H}}=1$. We have shown in the Abelian case
in Sec. II that gauging the U(1) symmetry has the same effect as condensing
$v$ in the original theory $\mathcal{C}$, and results in a gauged theory with
U(1)-charge-neutral excitations. In App. E, we show, using a combination of
field-theoretic and algebraic approaches, that the same is also true when we
gauge the U(1) symmetry of the SU$(N)_{k}$ MTC with a vanishing Hall
conductance. Physically, a vanishing Hall conductance $\sigma_{H}=0$ leads to
the absence of a CS term for the U(1) gauge field $A$. When the gauge field
$A$ becomes dynamical in the absence of any CS term, the instantons of the
gauge field $A$ is expected proliferate resulting in the condensation of
$2\pi$ flux which is tied to the vison $v\in\mathcal{C}$. Therefore, we expect
that, for a general U(1)-symmetric (2+1)d topological order described by the
MTC $\mathcal{C}$, gauging the U(1) symmetry of $\mathcal{C}$ has the same
effect as the condensing the vison $v$ in MTC $\mathcal{C}$. Such condensation
is permissible because of $\theta_{v}=1$ (a consequence of the vanishing Hall
conductance) and yields a new MTC $\mathcal{D}$ as the result of gauging the
U(1) symmetry of original theory $\mathcal{C}$.

Since condensing an Abelian anyon with a bosonic self-statistics does not
change the chiral central charge of the MTC, gauging the U(1) symmetry of
(2+1)d topological order with a vanishing Hall conductance leaves the chiral
central charge invariant.

##  V Dual U(1)dual symmetry after U(1) gauging

When gauging a finite group $G$, the gauged theory always has a subcategory
isomorphic to Rep$(G)$ (i.e. the symmetric tensor category of irreducible
linear representations of $G$). Condensing Rep$(G)$ performs “ungauging” and
returns to the original theory Barkeshli  _et al._ (2019); Tarantino  _et al._
(2016); Teo  _et al._ (2015); Lan  _et al._ (2017); Kong  _et al._ (2020).
When $G$ is a finite Abelian group, this phenomenon can be equivalently
formulated as the gauged theory having a (non-anomalous) 1-form symmetry group
isomorphic to $G$, and gauging the 1-form symmetry produces the ungauged
theory Gaiotto  _et al._ (2015); Hsin  _et al._ (2019). In other words,
gauging a finite Abelian (0-form) symmetry leads to a dual 1-form symmetry.
Notice that this statement holds independent of the actual low-energy dynamics
of the theory.

We now describe the analogy of dual symmetry in the case of U(1) gauging.
Let’s consider a U(1)-symmetric (2+1)d topological order described by the MTC
$\mathcal{C}$. We start by assuming a non-vanishing Hall conductance, i.e.
$\sigma_{H}\neq 0$. After gauging the U(1) symmetry of $\mathcal{C}$, we
obtain a new topological order whose corresponding MTC is $\mathcal{D}$. The
procedure in obtaining the categorical data of the MTC $\mathcal{D}$ is
discussed in Sec. III. In fact, the topological order $\mathcal{D}$ can admit
a dual U(1) symmetry, which we denote as U(1)dual. From the field theory
perspective, the current of this U(1)dual symmetry is given by $\star
dA/(2\pi)$ with $A$ the gauge field associated with the original U(1) symmetry
(before it is gauged) and $\star$ the Hodge star operator. Physically, the
U(1)${}_{\text{dual}}$ symmetry is the conservation of magnetic flux of the
original U(1) symmetry. When the original U(1) symmetry is gauged, $dA/(2\pi)$
becomes a dynamical object which can be minimally coupled to the (non-
dynamical) background U(1)dual gauge field $A^{\rm dual}$ via

$\int\frac{1}{2\pi}A^{\rm dual}dA.$ (32)

Since the dynamics of $A$ is effectively governed by the action Eq. (17), the
Hall conductance of the gauged theory $\mathcal{D}$ with respect to the
U(1)dual symmetry is given by

$\sigma_{H}^{\prime}=-1/\sigma_{H}.$ (33)

There is a new vison $v^{\prime}\in\mathcal{D}$ in the gauged theory
$\mathcal{D}$ that is associated with the $2\pi$ flux insertion of the $A^{\rm
dual}$. Eq. (32) implies that the vison $v^{\prime}$ should be identified just
as the unit charge of $A$, which further corresponds to the anyon $(1,1)$ in
the infinite category $\mathcal{C}^{\prime}$ introduced in Sec. III. As
discussed above, the gauged theory $\mathcal{D}$ can be obtained from the
infinite category $\mathcal{C}^{\prime}$ via the condensation of the
transparent anyon $(v,\sigma_{H})\in\mathcal{C}^{\prime}$. Since this
condensation does not change the topological twist factors, we can obtain the
topological twist factor of the vison $v^{\prime}\in\mathcal{D}$ using Eq.
(20):

$\displaystyle\theta_{v^{\prime}}=\theta_{(1,1)}=e^{-\mathrm{i}\pi/\sigma_{H}}.$
(34)

Notice that the consistency condition
$\theta_{v^{\prime}}=e^{\mathrm{i}\pi\sigma_{H}^{\prime}}$ is satisfied in the
gauged theory $\mathcal{D}$ with the U(1)dual symmetry. If one consider
further gauging the U(1)dual symmetry of the theory $\mathcal{D}$, it is
obvious from the field theory perspective that resulting theory should be
identical to the original theory $\mathcal{C}$ with the same U(1) symmetry
enrichment, namely the same Hall conductance $\sigma_{H}$ and same vison
$v\in\mathcal{C}$, as before.

It can happen that in going from $\mathcal{C}^{\prime}$ to $\mathcal{D}$,
$(1,1)$ is also condensed when it is generated by $(v,\sigma_{H})\equiv
T_{1}$. In other words, if this happens we must have
$(1,1)=(v^{s},s\sigma_{H})$, which implies $\sigma_{H}=\frac{1}{s}$. In this
case, since $\theta_{v}=e^{\mathrm{i}\pi/s}$, the subcategory generated by $v$
is identified with $\mathrm{U}(1)_{s}$, which is an MTC on its own. By the
factorization property of MTCs, it implies that the original MTC $\mathcal{C}$
takes the form $\mathcal{C}=\mathcal{D}\boxtimes\mathrm{U}(1)_{s}$, and it is
not difficult to see that $\mathcal{D}$ is indeed the gauged theory (hence the
notation). In this case the dual vison is $v^{\prime}=1$.

When $\sigma_{H}=0$, the most relevant term of the gauge field $A$ generated
by the “matter fields” in the original theory $\mathcal{C}$ is a (2+1)d
Maxwell term. When the original U(1) symmetry is gauged, the gauge field $A$
is governed by a Maxwell theory, whose flux is now conserved because of the
U(1)dual symmetry. The Polyakov’s instanton proliferation mechanism is
forbidden by the U(1)dual symmetry. In this case, assuming that the matter
fields of the original theory $\mathcal{C}$ remains gapped, the resulting
phase is a gapless phase whose low-energy modes are given by the deconfined
and gapless photons of the gauge field $A$. In fact, in this phase, the
U(1)dual symmetry is spontaneously broken and the corresponding Goldstone
modes are dual to the gapless photons of the gauge field $A$. In this gapless
phase, it is no longer appropriate to characterize the resulting phase of
matter using just MTCs. Note that if there is no U(1)dual symmetry presence,
the confinement of the gauge field $A$ through the instanton proliferation
mechanism should always happen resulting in a gapped phase as the gauged
theory. And the resulting topological order follows from the discussion of
Sec. IV.

In principle, one can also consider the scenario when the gauge field $A$
becomes higgsed. In this scenario, U(1)dual symmetry is no longer
spontaneously broken and the resulting phase should be gapped. The topological
order of this U(1)dual-symmetric gapped phase will depend on the choice of the
Higgs field. The most trivial situation is when the Higgs field is
topologically equivalent to the trivial anyon in the original theory
$\mathcal{C}$ before the original U(1) symmetry is gauged. The resulting
topological order is still given by an MTC $\mathcal{C}$. But its enrichment
under the U(1)dual symmetry is trivial.

##  VI Discussion

MTCs from Chern-Simons theory of compact Lie groups are closely related to
Wess-Zumino-Witten (WZW) chiral conformal field theories in (1+1)d. It is
known that gauging a subgroup in a WZW theory is equivalent to the coset
construction of the WZW CFT Nahm (1987); Gawedzki and Kupiainen (1989);
Bardakci  _et al._ (1988). For example, $\mathbb{Z}_{k}$ parafermion CFT can
be viewed as the coset SU(2)${}_{k}/\mathrm{U}(1)$ theory. Thus the gauging
prescription we gave is basically the categorical version of coset by U(1).
Since coset construction works for any Lie group symmetry, it is an important
question to develop a categorical description of gauging for general compact
Lie groups. We address this question in a follow-up publication Cheng  _et
al._ (2022b)

In the finite group case, Ref. [Barkeshli  _et al._ , 2019] describes the
gauging in two steps: first symmetry defects are introduced and together with
anyons they form a mathematical structure called $G$-crossed braided tensor
category. Then an “equivariantization” procedure is applied to obtained an MTC
corresponding to the gauged system, which is physically the projection to
$G$-invariant subspace. Formally one can also define “U(1)-crossed” braided
category, as shown in Ref. [Barkeshli  _et al._ , 2019] and Ref. [Cheng  _et
al._ , 2016; Manjunath and Barkeshli, 2020; Barkeshli, ], where symmetry
defects are labeled by elements of U(1). It will be interesting to understand
how this approach to gauging is related to ours.

Another direction for future work is to generalize the construction to
fermionic systems with $\mathrm{U}(1)_{f}$ symmetry, where $\mathrm{U}(1)_{f}$
is the conservation of fermion number. In other words, local excitations with
odd/even charge are fermions/bosons. We expect that the basic strategy in this
work can be generalized, but there may be additional sign factors in the $F$\-
and $R$-symbols for the category $\mathcal{C}^{\prime}$ coming from Fermi
statistics.

Moreover, a recent work Ref. [Kobayashi and Barkeshli, 2021] provides a
general analysis on the coupling between (2+1)d topological orders and general
curved U(1) background gauge fields. Establishing the relation between the
analysis in Ref. [Kobayashi and Barkeshli, 2021] and our general framework for
U(1) symmetry gauging (where the U(1) gauge field becomes dynamical) will be
left for future work.

##  VII Acknowledgment

We thank M. Barkeshli for enlightening conversations and sharing unpublished
results, and P. Bonderson and T. Lan for feedbacks on a draft of the
manuscript. We are especially grateful to P. Bonderson for explaining the
hierarchy construction for fractional quantum Hall states. C.-M.J. thanks D.
Aasen for helpful discussions. M.C. acknowledges support from NSF under award
number DMR-1846109.

##  Appendix A Minimal charge

The excitation $v$ induced by a $2\pi$ flux must be an Abelian anyon. Suppose
$v$ has $\mathbb{Z}_{s}$ fusion rule in the bosonic MTC (before gauging U(1)),
let’s show that the minimal charge $e^{*}$ has to be $\frac{1}{s}$. For any
anyon $a$, since $e^{2\pi\mathrm{i}Q_{a}}=M_{av}$, the $\mathbb{Z}_{s}$ fusion
rule of $v$ requires $sQ_{a}\in\mathbb{Z}$. All possible charges that can by
any anyonic or local excitation in this MTC is given by
$(\oplus_{a}Q_{a}\mathbb{Z})\oplus\mathbb{Z}$. Our goal is to show that
$\frac{1}{s}\in(\oplus_{a}Q_{a}\mathbb{Z})\oplus\mathbb{Z}$.

Let’s prove it by contraction. Let’s assume
$\frac{1}{s}\notin(\oplus_{a}Q_{a}\mathbb{Z})\oplus\mathbb{Z}$. It implies
that greatest common divisor $n$ of the set of integers
$\\{sQ_{a}\\}_{a}\cup\\{s\\}$ is greater than 1, i.e. $n>1$. Then, we can
consider an non-trivial anyon $x$ which is the fusion product of $\frac{s}{n}$
anyons $v$. $x$ is an non-trivial anyon because $v$ has $\mathbb{Z}_{s}$
fusion rules. Notice that
$M_{xa}=(M_{va})^{\frac{s}{n}}=e^{2\pi\mathrm{i}Q_{a}s/n}=1$ for all anyon
$a$. However, in an MTC, the only “anyon” with this property is the trivial
anyon. Hence, we arrive at a contradiction. And therefore, we prove that
$\frac{1}{s}\in(\oplus_{a}Q_{a}\mathbb{Z})\oplus\mathbb{Z}$, namely
$e^{*}=\frac{1}{s}$. This statement further implies that
$(0,\sigma_{H}/e^{*})$ is always generated by $(v,Q_{v})$.

##  Appendix B Condensing transparent anyons

###  B.1 General Formalism

As an intermediate step in gauging the U(1) symmetry of the MTC $\mathcal{C}$,
we have introduced in Sec. III the infinite category $\mathcal{C}^{\prime}$
are contains all possible excitations $(a,Q_{a})$. The fusion rule of this
infinite category $\mathcal{C}^{\prime}$ is given by Eq. (18) and the $F$\- an
$R$-symbols are given by Eq. (21). For simplicity, we will only focus on the
situation where all of the fusion multiplicity $N^{c}_{ab}$ of the original
MTC $\mathcal{C}$ before gauging U(1) is less than or equal to 1, i.e.
$N^{c}_{ab}\leq 1$. In this appendix, we use a a single greek letter to denote
the pair $(a,Q_{a})$, for example $\alpha=(a,Q_{a})$, to simplify the
notation. The assumption that $N^{c}_{ab}\leq 1$ in the original theory before
gauging U(1) implies that fusion multiplicity $N^{\gamma}_{\alpha\beta}$ of
the infinite category $\mathcal{C}^{\prime}$ is also less than or equal to 1.

We are interested in the category $\tilde{\mathcal{D}}$ obtained from
condensing a group of transparent Abelian anyons $\mathcal{T}$ in the infinite
category $\mathcal{C}^{\prime}$. In particular, we focus on the group
$\mathcal{T}=\\{\tau^{k}\\}_{k\in\mathbb{Z}}$ generated by a single
transparent Abelian anyon $\tau$. As discussed in the main text, for the
purpose of U(1) gauging, we should condense the group of transparent Abelian
anyons $\mathcal{T}$ generated by $\tau=(v,\sigma_{H})$. In this case, the
resulting category $\tilde{\mathcal{D}}$ is the category $\mathcal{D}$ that is
the final result of gauging the U(1) symmetry in the MTC $\mathcal{C}$. As we
will see later, it is also helpful to consider condensing the group generated
by $\tau=(1,s\sigma_{H})$ (which is the fusion product of $s$ copies of
$(v,\sigma_{H})$). In this case, the category $\tilde{\mathcal{D}}$ is another
intermediate premodular category (with finitely many anyons) towards the final
gauged theory $\mathcal{D}$. The following discussion of the condensation of
transparent Abelian anyons will be applicable to both cases unless specified
otherwise.

The anyons in $\mathcal{C}^{\prime}$ form orbits under the fusion with
$\mathcal{T}$. Let’s label these orbits by $[\alpha]$. For each orbit
$[\alpha]$, we pick a representative $\alpha\in[\alpha]$. The orbit can be
then expressed as $[\alpha]=\\{\alpha\tau^{k}\\}_{k\in\mathbb{Z}}$. Since
$\tau^{k}$ for different $k$’s carry different U(1) charges,
$\alpha\tau^{k}\in[\alpha]$ with different $k$ are different anyons. This
property has an important consequence that, when we condense the transparent
Abelian anyons $\mathcal{T}\subset\mathcal{C}^{\prime}$ to obtain the category
$\tilde{\mathcal{D}}$, the orbits $[\alpha]$ are in one-to-one correspondence
to the anyons types in $\tilde{\mathcal{D}}$. Therefore, we will directly use
the orbit labels $[\alpha]$ to denote the anyons in $\tilde{\mathcal{D}}$. The
fusion rule in $\tilde{\mathcal{D}}$ can be directly obtained from that of the
parent category $\mathcal{C}^{\prime}$:

$\displaystyle N^{[\gamma]}_{[\alpha][\beta]}=\begin{cases}1,~{}~{}~{}\text{if
there exists $\Delta^{[\gamma]}_{[\alpha][\beta]}\in\mathcal{T}$ such that
$N^{\gamma\left(\Delta^{[\gamma]}_{[\alpha][\beta]}\right)^{-1}}_{\alpha\beta}=1$
in $\mathcal{C}^{\prime}$, }\\\ 0,~{}~{}~{}\text{otherwise.}\end{cases}$ (35)

Remember the fusion multiplicity in $\mathcal{C}^{\prime}$ is assumed to be
equal to or less than 1. Physically, this fusion rule of $\tilde{\mathcal{D}}$
means that $[\alpha]$ and $[\beta]$ can fuse into $[\gamma]$ so long as their
representatives $\alpha$ and $\beta$ can fuse into the representative $\gamma$
up to some condensed transparent anyon
$\Delta^{[\gamma]}_{[\alpha][\beta]}\in\mathcal{T}$.
$\Delta^{[\gamma]}_{[\alpha][\beta]}$ depends on the choice of representatives
of each orbit $[\alpha]$, $[\beta]$ and $[\gamma]$.
$\Delta^{[\gamma]}_{[\alpha][\beta]}$ is symmetric under the exchange of its
two lower indices, i.e.
$\Delta^{[\gamma]}_{[\alpha][\beta]}=\Delta^{[\gamma]}_{[\beta][\alpha]}$.

Now, we calculate the $F$\- and $R$-symbols of the category
$\tilde{\mathcal{D}}$. As a first step, it is convenient to choose a gauge of
the $F$\- and $R$-symbols of $\mathcal{C}^{\prime}$ such that they all take
value $1$ when restricted to the group of transparent anyons $\mathcal{T}$.
Following the discussion of App. C, such a gauge always exists for
$\mathcal{T}$ which forms the group $\mathbb{Z}$ under fusion. Let’s denote
the $F$\- and $R$-symbols of $\mathcal{C}^{\prime}$ in the desired gauge as
$F^{\prime}$ and $R^{\prime}$. Note that when $\tau=(1,s\sigma_{H})$, the
$F$\- and $R$-symbols in Eq. (21) are already in the desired gauge. However,
this is generically not the case when $\tau=(v,\sigma_{H})$. Hence, additional
gauge transformation to Eq. (21) is needed to obtain $F^{\prime}$ and
$R^{\prime}$ in the desired gauge in the case of $\tau=(v,\sigma_{H})$.

The anyon diagrams of the category $\tilde{\mathcal{D}}$ can be lifted back to
the anyon diagrams of the parent infinite category $\mathcal{C}^{\prime}$. In
particular, the fusion vertex involving anyons $[\alpha]$, $[\beta]$ and
$[\gamma]$ in $\tilde{\mathcal{D}}$ can be expressed as the fusion diagram of
$\mathcal{C}^{\prime}$ shown in Fig. 1 (a). We use thick black line for the
anyons in $\tilde{\mathcal{D}}$, thin black lines for anyons in
$\mathcal{C}^{\prime}$ and blue lines for condensed transparent Abelian anyon
in $\mathcal{T}\subset\mathcal{C}^{\prime}$ with the blue dots denoting where
the condensation occur. When a generic diagram of $\tilde{\mathcal{D}}$ is
lifted to $\mathcal{C}^{\prime}$, we always adopt the conventions that the all
blue lines for the condensed transparent Abelian anyons stay underneath all
the black anyon lines and they all condense at the left most part of the
diagram. In principle, we should also consider fusing all of the condensed
transparent Abelian anyon before their condensation. Due to the gauge choice
of $F^{\prime}$ and $R^{\prime}$ which are trivial when restricted to the
group $\mathcal{T}$ of condensed transparent Abelian anyons, these transparent
Abelian anyons can separately condense without extra phase factor associated
with the locations of their condensation. Also, this gauge choice allows us to
split, without introducing extra phase factor, a condensed anyon line of
$\tau^{k_{1}+k_{2}}\in\mathcal{T}$ in the way shown in Fig. 1 (b) for any
$k_{1,2}\in\mathbb{Z}$.

Figure 1: (a) The fusion vertex of the category $\tilde{\mathcal{D}}$ can be
lifted into the parent theory $\mathcal{C}^{\prime}$ as a fusion diagram shown
here. (b) For any $k_{1,2}\in\mathbb{Z}$, a single line of condensed
transparent anyon $\tau^{k_{1}+k_{2}}$ can be split without extra phase given
the proper gauge choice of the $F$\- and $R$-symbols in $\mathcal{C}^{\prime}$
as explained in the main text.

The $F$-symbol of the category $\tilde{\mathcal{D}}$ can be calculated using
the parent category $\mathcal{C}^{\prime}$. As is shown in Fig. 2, a single
$F$-move in the category $\tilde{\mathcal{D}}$ when lifted to the parent
category $\mathcal{C}^{\prime}$ consists of a sequence of $F$\- and $R$-moves
in $\mathcal{C}^{\prime}$. Hence, the $F$-symbol
$\left(F^{[\alpha],[\beta],[\gamma]}_{[\delta]}\right)_{[\varepsilon],[\varphi]}$
of the category $\tilde{\mathcal{D}}$ can be written in terms of $F^{\prime}$
and $R^{\prime}$ of the category $\mathcal{C}^{\prime}$

$\displaystyle\left(F^{[\alpha],[\beta],[\gamma]}_{[\delta]}\right)_{[\varepsilon],[\varphi]}=$
$\displaystyle\left({F^{\prime}}^{\Delta_{[\alpha][\beta]}^{[\varepsilon]},\varepsilon^{\prime},\gamma}_{\delta^{\prime}}\right)_{\varepsilon,\delta^{\prime\prime}}\left({F^{\prime}}^{\alpha,\beta,\gamma}_{\delta^{\prime\prime}}\right)_{\varepsilon^{\prime},\varphi^{\prime}}\left({F^{\prime}}^{\Delta_{[\varepsilon][\gamma]}^{[\delta]},\Delta_{[\alpha][\beta]}^{[\varepsilon]},\delta^{\prime\prime}}_{\delta}\right)^{-1}\left({F^{\prime}}^{\Delta_{[\alpha][\varphi]}^{[\delta]},\Delta_{[\beta][\gamma]}^{[\varphi]},\delta^{\prime\prime}}_{\delta}\right)$
$\displaystyle\times\left({F^{\prime}}^{\Delta_{[\beta][\gamma]}^{[\varphi]},\alpha,\varphi^{\prime}}_{\delta^{\prime\prime\prime}}\right)^{-1}\left({R^{\prime}}^{\Delta_{[\beta][\gamma]}^{[\varphi]},\alpha}\right)^{-1}\left({F^{\prime}}^{\alpha,\Delta_{[\beta][\gamma]}^{[\varphi]},\varphi^{\prime}}_{\delta^{\prime\prime\prime}}\right),$
(36)

where $\alpha,\beta,\gamma,\varepsilon$ and $\varphi$ are the representatives
of their corresponding orbits $[\alpha],[\beta],[\gamma],[\varepsilon]$ and
$[\varphi]$. Also, we’ve defined the following anyon variables

$\displaystyle\alpha^{\prime}=\Delta_{[\beta][\gamma]}^{[\varphi]}\alpha,~{}~{}~{}~{}\varepsilon^{\prime}=\left(\Delta_{[\alpha][\beta]}^{[\varepsilon]}\right)^{-1}\varepsilon,~{}~{}~{}~{}\varphi^{\prime}=\left(\Delta_{[\beta][\gamma]}^{[\varphi]}\right)^{-1}\varphi,$
$\displaystyle\delta^{\prime}=\left(\Delta_{[\varepsilon][\gamma]}^{[\delta]}\right)^{-1}\delta,~{}~{}~{}~{}\delta^{\prime\prime}=\left(\Delta_{[\alpha][\beta]}^{[\varepsilon]}\right)^{-1}\delta^{\prime},~{}~{}~{}~{}\delta^{\prime\prime\prime}=\left(\Delta_{[\alpha][\varphi]}^{[\delta]}\right)^{-1}\delta.$
(37)

On the second row of Fig. 2, we’ve used the fact that
$\Delta_{[\alpha][\beta]}^{[\varepsilon]}\Delta_{[\varepsilon][\gamma]}^{[\delta]}=\Delta_{[\alpha][\varphi]}^{[\delta]}\Delta_{[\beta][\gamma]}^{[\varphi]}$
are the same transparent Abelian anyon. We’ve suppressed certain indices of
the $F$\- and $R$-symbol to avoid cluttering in Eq. 36. All the suppressed
indices are fully determined via the fusion rule by the explicitly written
indices in the same $F$\- or $R$-symbols.

Figure 2: A single $F$-move in the cateory $\tilde{\mathcal{D}}$, when lifted
to the parent category $\mathcal{C}^{\prime}$, consists of a sequence of $F$\-
and $R$-moves in $\mathcal{C}^{\prime}$

The $R$-symbol of the category $\tilde{\mathcal{D}}$ can also be obtained by
lifting the associated anyon diagram to the parent category
$\mathcal{C}^{\prime}$, as shown in Fig. 3:

$\displaystyle
R^{[\alpha],[\beta]}_{[\gamma]}={R^{\prime}}^{\alpha,\beta}_{\gamma^{\prime}},$
(38)

where
$\gamma^{\prime}=\left(\Delta_{[\alpha][\beta]}^{[\gamma]}\right)^{-1}\gamma$.

Figure 3: A $R$-move in the category $\tilde{\mathcal{D}}$ can be lifted to a
$R$-move in the parent category $\mathcal{C}^{\prime}$

We note that similar diagrammatics has been used to compute $F$\- and
$R$-symbols of symmetry defects in a $G$-crossed braided tensor category
Barkeshli and Cheng (2020); Aasen  _et al._ (2021).

###  B.2 Condensation of $\tau=(1,s\sigma_{H})$

Now, we consider the case in which the group $\mathcal{T}$ of transparent
Abelian anyons is generated by $\tau=(1,s\sigma_{H})$. The category obtained
from condensing this group $\mathcal{T}$ in $\mathcal{C}^{\prime}$ will be
denoted by $\mathcal{D}_{\rm int}$. As mentioned earlier, the category
$\mathcal{D}_{\rm int}$ is an intermediate premodular category towards the
final theory $\mathcal{D}$ where the U(1) symmetry of the original theory
$\mathcal{C}$ is fully gauged.

As commented above, the $F$\- and $R$-symbols of $\mathcal{C}^{\prime}$ given
in Eq. (21) is already in the desired gauge, namely their restrictions to the
group $\mathcal{T}$ are completely trivial. Under the fusion with transparent
anyons in $\mathcal{T}$, each of the orbits in $\mathcal{C}^{\prime}$ take the
form $[(a,Q_{a})]=\\{(a,Q_{a}+ks\sigma_{H})\\}_{k\in\mathbb{Z}}$. Hence, for
each orbit $[(a,Q_{a})]$, we can choose $(a,\lceil
Q_{a}\rfloor_{|s\sigma_{H}|})\in[(a,Q_{a})]$ as its representative where
$\lceil Q_{a}\rfloor_{|s\sigma_{H}|}$ denotes the residue of any $Q_{a}$ that
appears in $[(a,Q_{a})]$ modulo $|s\sigma_{H}|$. $\lceil
Q_{a}\rfloor_{|s\sigma_{H}|}$ takes value within the interval
$[0,|s\sigma_{H}|)$. When $N_{[(a,Q_{a})][(b,Q_{b})]}^{[(c,Q_{c})]}\neq 0$,
U(1) charge conservation together with the fact that $\lceil
Q_{a}\rfloor_{|s\sigma_{H}|},\lceil Q_{b}\rfloor_{|s\sigma_{H}|},\lceil
Q_{c}\rfloor_{|s\sigma_{H}|}\in[0,s\sigma_{H})$ leads to the consequence that

$\displaystyle\Delta_{[(a,Q_{a})][(b,Q_{b})]}^{[(c,Q_{c})]}=(1,\lceil
Q_{a}+Q_{b}\rfloor_{|s\sigma_{H}|}-\lceil Q_{a}\rfloor_{|s\sigma_{H}|}-\lceil
Q_{b}\rfloor_{|s\sigma_{H}|}).$ (39)

Now, following App. B.1, we can calculate the $F$\- and $R$-symbols of the
category $\mathcal{D}_{\rm int}$ obtained from condensing the group of
transparent Abelian anyons $\mathcal{T}$ in the parent category
$\mathcal{C}^{\prime}$. Notice that, the $F$-symbol
$\left(F^{(a,Q_{a}),(b,Q_{b}),(c,Q_{c})}_{(d,Q_{d})}\right)_{(e,Q_{e}),(f,Q_{f})}$
of the category $\mathcal{C}^{\prime}$ shown in Eq. (21) is $1$ when any one
of $(a,Q_{a})$, $(b,Q_{b})$, and $(c,Q_{c})$ belongs to $\mathcal{T}$. This
observation greatly simplifies the expression Eq. (36) and results

$\displaystyle\left(F^{[(a,Q_{a})],[(b,Q_{b})],[(c,Q_{c})]}_{[(d,Q_{d})]}\right)_{[(e,Q_{e})],[(f,Q_{f})]}=\left(F^{a,b,c}_{d}\right)_{e,f}e^{\frac{\pi\mathrm{i}}{\sigma_{H}}\lceil
Q_{a}\rfloor_{|s\sigma_{H}|}(\lceil Q_{b}+Q_{c}\rfloor_{|s\sigma_{H}|}-\lceil
Q_{b}\rfloor_{|s\sigma_{H}|}-\lceil Q_{c}\rfloor_{|s\sigma_{H}|})}$ (40)

The $R$-symbol of $\mathcal{D}_{\rm int}$ can be obtained via Eq. (38):

$\displaystyle
R^{[(a,Q_{a})],[(b,Q_{b})]}_{[(c,Q_{c})]}=R^{ab}_{c}e^{-\frac{\pi\mathrm{i}}{\sigma_{H}}\lceil
Q_{a}\rfloor_{|s\sigma_{H}|}\lceil Q_{b}\rfloor_{|s\sigma_{H}|}}.$ (41)

Here, as a reminder, the $F$-symbol $\left(F^{a,b,c}_{d}\right)_{e,f}$ and the
$R$-symbol $R^{ab}_{c}$ are those of the original category $\mathcal{C}$.

It is easy to see that the category $\mathcal{D}_{\rm int}$ is still a
premodular category where the anyon represented by $[(v,\sigma_{H})]$ is a
transparent Abelian anyon. After we further condense the transparent Abelian
anyon $[(v,\sigma_{H})]$ (and the ones generated by it) in $\mathcal{D}_{\rm
int}$, the resulting category is the category $\mathcal{D}$ that is the final
outcome of gauging the U(1) symmetry of the original category $\mathcal{C}$.

###  B.3 Application to SU(2)k

Consider a 2+1d U(1)-symmetric topological phase whose topological order is
given by the MTC $\mathcal{C}=\text{SU}(2)_{k}$. There are $k+1$ different
types of anyons, labeled by $j=0,1/2,\cdots,k/2$. There is a single nontrivial
Abelian anyon with $j=k/2$. The fusion rules are given by

$j_{1}\times j_{2}=|j_{1}-j_{2}|+\cdots+\min(j_{1}+j_{2},k-j_{1}-j_{2}).$ (42)

In particular, $\frac{k}{2}\times j=\frac{k}{2}-j$. The Abelian anyon $k/2$
forms a $\mathbb{Z}_{2}$ group under fusion. The fusion multiplicities of
SU(2)k are given by

$\displaystyle
N_{j_{1}j_{2}}^{j}=\begin{cases}1,~{}~{}j\in\\{|j_{1}-j_{2}|,~{}|j_{1}-j_{2}|+1,~{}...,~{}\min(j_{1}+j_{2},k-j_{1}-j_{2})\\},\\\
0,~{}~{}\text{otherwise.}\end{cases}$ (43)

The $F$-symbols of SU(2)k are given by Kirillov and Reshetikhin (1988)

$\displaystyle\left(F^{j_{1},j_{2},j_{3}}_{j}\right)_{j_{12},j_{23}}=(-1)^{j_{1}+j_{2}+j_{3}+j}\sqrt{\\{2j_{12}+1\\}_{q}\\{2j_{23}+1\\}_{q}}\left[\begin{array}[]{ccc}j_{1}&j_{2}&j_{12}\\\
j_{3}&j&j_{23}\end{array}\right]_{q}$ (46)

where $q=e^{\mathrm{i}\frac{2\pi}{k+2}}$,
$\\{n\\}_{q}\equiv\frac{q^{n/2}-q^{-n/2}}{q^{1/2}-q^{-1/2}}$ and

$\displaystyle\left[\begin{array}[]{ccc}j_{1}&j_{2}&j_{12}\\\
j_{3}&j&j_{23}\end{array}\right]_{q}=$
$\displaystyle\Upsilon(j_{1},j_{2},j_{12})\Upsilon(j_{12},j_{3},j)\Upsilon(j_{2},j_{3},j_{23})\Upsilon(j_{1},j_{23},j)$
(49)
$\displaystyle\times\sum_{z}\left[\frac{(-1)^{z}\\{z+1\\}_{q}!}{\\{z-j_{1}-j_{2}-j_{12}\\}_{q}!\\{z-j_{12}-j_{3}-j\\}_{q}!\\{z-j_{2}-j_{3}-j_{23}\\}_{q}!\\{z-j_{1}-j_{23}-j\\}_{q}!}\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\times\frac{1}{\\{j_{1}+j_{2}+j_{3}+j-z\\}_{q}!\\{j_{1}+j_{12}+j_{3}+j_{23}-z\\}_{q}!\\{j_{2}+j_{12}+j+j_{23}-z\\}_{q}!}\right]$
(50)

We’ve used the following definition in Eq. (50),

$\displaystyle\\{n\\}_{q}!\equiv\prod_{m=1}^{n}\\{m\\}_{q},$
$\displaystyle\Upsilon(j_{1},j_{2},j_{12})\equiv\sqrt{\frac{\\{-j_{1}+j_{2}+j_{3}\\}_{q}!\\{j_{1}-j_{2}+j_{3}\\}_{q}!\\{j_{1}+j_{2}-j_{3}\\}_{q}!}{\\{j_{1}+j_{2}+j_{3}+1\\}_{q}!}}.$
(51)

The summation $\sum_{z}$ in Eq. (50) runs over all the integer values of $z$
such that the arguments of all $\\{\cdot\\}_{q}!$ functions that appear are
non-negative.

The $R$-symbol of SU(2)k is given by

$\displaystyle
R^{j_{1},j_{2}}_{j}=(-1)^{j-j_{1}-j_{2}}q^{\frac{1}{2}(j(j+1)-j_{1}(j_{1}+1)-j_{2}(j_{2}+1))}.$
(52)

Since the anyon $j=k/2$ is the only Abelian anyon in SU(2)k, it should also be
identified as the anyon $v$ associated with the $2\pi$ flux (or the vison),
i.e. $v=k/2$. Since $\theta_{k/2}=e^{\pi ik/2}$, the Hall conductance
satisfies $\sigma_{H}=\frac{k}{2}$ mod $2\mathbb{Z}$.

When $k$ is odd, the $k/2$ anyon is a semion (or anti-semion) and the whole
MTC factorizes into a “SO(3)k” MTC and the semion (or anti-semion) theory. The
U(1) symmetry only acts non-trivially on the semion (or anti-semion) sector.
Therefore, the gauging of the U(1) symmetry only alters the semion (or anti-
semion) sector without changing the “SO(3)k” sector.

The case with even $k$ is more interesting. We will focus on this case in the
following. Following the general prescription, we need to introduce the
infinite category $\mathcal{C}^{\prime}$ whose anyons are labeled by $(j,Q)$
where $j$ labels the anyon in SU(2)k and $Q$ labels the U(1) charge. Since the
anyon that corresponds to the $2\pi$ flux is given by $v=k/2$, the U(1) charge
of $(j,Q)$ satisfies the constraint that
$e^{2\pi\mathrm{i}j}=e^{2\pi\mathrm{i}Q}$. The $F$\- and $R$-symbols of the
category $\mathcal{C}^{\prime}$ can be obtained using Eq. (21), 46 and 52:

$\displaystyle\left(F^{\prime(j_{1},Q_{1}),(j_{2},Q_{2}),(j_{3},Q_{3})}_{(j,Q)}\right)_{(j_{12},Q_{12}),(j_{23},Q_{23})}=\left(F^{j_{1},j_{2},j_{3}}_{j}\right)_{j_{12},j_{23}},$
$\displaystyle
R^{\prime(j_{1},Q_{1}),(j_{2},Q_{2})}_{(j,Q)}=R^{j_{1},j_{2}}_{j}e^{-\frac{\pi\mathrm{i}}{\sigma_{H}}Q_{1}Q_{2}}$
(53)

To obtain the MTC $\mathcal{D}$, i.e. the final result of gauging the U(1)
symmetry of $\mathcal{C}$, the group $\mathcal{T}$ of transparent Abelian
anyon to be condensed is generated by $\tau=(\frac{k}{2},\sigma_{H})$. Note
that, $(\frac{k}{2},-\sigma_{H})\in\mathcal{T}$ since the anyon $k/2$ has a
$\mathbb{Z}_{2}$-fusion rule. The anyons in $\mathcal{C}^{\prime}$ form orbit
under fusion with $\mathcal{T}$. One can show that the orbits are completely
labeled by $[(j,Q)]$ for $j=0,1,2,...,k/2$ and $0\leq Q<|\sigma_{H}|$. We will
use $(j,Q)$ within the same range of $j$ and $Q$ as the representative of its
corresponding orbit $[(j,Q)]$. As discussed above, each orbit $[(j,Q)]$ also
labels an anyon in the category $\mathcal{D}$ obtained from condensing the
group of transparent anyons $\mathcal{T}$.

In the following, we will obtain the data that defines the category
$\mathcal{D}$. It is convenient to introduce the function

$\displaystyle\Lambda(Q_{1},Q_{2})=\frac{1}{\sigma_{H}}(Q_{1}+Q_{2}-\lceil
Q_{1}+Q_{2}\rfloor_{|\sigma_{H}|}),$ (54)

where $\lceil Q\rfloor_{|\sigma_{H}|}$ is the residue of $Q$ modulo
$|\sigma_{H}|$ and it satisfies $0\leq\lceil
Q\rfloor_{|\sigma_{H}|}<|\sigma_{H}|$. For any $0\leq
Q_{1},Q_{2}<|\sigma_{H}|$, the function $\Lambda(Q_{1},Q_{2})$ only takes
value 0 or $\pm 1$. The fusion rule of category $\mathcal{D}$ is given by

$[(j_{1},Q_{1})]\times[(j_{2},Q_{2})]=\begin{cases}\sum_{j}N_{j_{1}j_{2}}^{j_{3}}(j,\lceil
Q_{1}+Q_{2}\rfloor_{|\sigma_{H}|}),~{}~{}~{}~{}&\text{for~{}}\Lambda(Q_{1},Q_{2})=0\\\
\\\ \sum_{j_{3}}N_{j_{1}j_{2}}^{j}(\frac{k}{2}-j,\lceil
Q_{1}+Q_{2}\rfloor_{|\sigma_{H}|}),~{}~{}~{}~{}&\text{for~{}}|\Lambda(Q_{1},Q_{2})|=1\end{cases}$
(55)

This fusion rule and our choice of orbit representatives lead to the function

$\Delta_{[(j_{1},Q_{1})][(j_{2},Q_{2})]}^{[(j,Q)]}=\left(\frac{k}{2},\sigma_{H}\right)^{-\Lambda(Q_{1},Q_{2})}$
(56)

when the fusion channel
$[(j_{1},Q_{1})]\times[(j_{2},Q_{2})]\rightarrow[(j,Q)]$ exists.

To obtain the $F$-symbol of $\mathcal{D}$, as explained in App. B.1, it is
better to start with the preferred gauge choice for the $F$\- and $R$-symbols
of $\mathcal{C}^{\prime}$ such that the $F$\- and $R$-symbols restricted to
$\mathcal{T}\in\mathcal{C}^{\prime}$ are trivial. Remember that we are
currently focusing on the case with even $k$. It turns out that the $F$\- and
$R$-symbols of $\mathcal{C}$’ obtained in Eq. (53) are already in the
preferred gauge choice when $k$ is even. This is due to the fact that
$\left(F^{k/2,k/2,k/2}_{k/2}\right)_{0,0}=1$ for even $k$ according to Eq.
(46). Hence, we can proceed to calculate the $F$-symbol of the category
$\mathcal{D}$ using Eq. (36) without any extra gauge transformation to Eq.
(53) needed. Additional simplification can be achieved by noticing that the
$F$-symbol of SU(2)k given in Eq. (46) has the properties

$\displaystyle\left(F^{j_{1}=\frac{k}{2},j_{2},j_{3}}_{j}\right)_{\frac{k}{2}-j_{2},\frac{k}{2}-j}=(-1)^{\frac{k}{2}-j_{2}-j_{3}-j},$
$\displaystyle\left(F^{j_{1},j_{2}=\frac{k}{2},j_{3}}_{j}\right)_{\frac{k}{2}-j_{1},\frac{k}{2}-j_{3}}=(-1)^{\frac{k}{2}-j_{1}-j_{3}-j},$
(57)
$\displaystyle\left(F^{j_{1},j_{2},j_{3}=\frac{k}{2}}_{j}\right)_{\frac{k}{2}-j,\frac{k}{2}-j_{2}}=(-1)^{\frac{k}{2}-j_{1}-j_{2}-j}.$

With these simplifications taken into account, the $F$-symbol of the MTC
$\mathcal{D}$ is given by

$\displaystyle\left(F^{[(j_{1},Q_{1})],[(j_{2},Q_{2})],[(j_{3},Q_{3})]}_{[(j,Q)]}\right)_{[(j_{12},Q_{12})],[(j_{23},Q_{23})]}$
$\displaystyle=(-1)^{\left(j_{12}-j_{3}-j^{\prime}\right)\Lambda(Q_{1},Q_{2})}(-1)^{2j\left(\Lambda(Q_{1},Q_{2})\Lambda(Q_{12},Q_{3})+\Lambda(Q_{1},Q_{23})\Lambda(Q_{2},Q_{3})\right)}(-1)^{(j_{1}\operatorname{sgn}\sigma_{H}+Q_{1})\Lambda(Q_{2},Q_{3})}\left(F^{j_{1},j_{2},j_{3}}_{j^{\prime\prime}}\right)_{j_{12}^{\prime},j_{23}^{\prime}},$
(58)

with

$\displaystyle
j^{\prime}=(1-|\Lambda(Q_{12},Q_{3})|)j+|\Lambda(Q_{12},Q_{3})|\left(\frac{k}{2}-j\right),$
$\displaystyle
j^{\prime\prime}=(1-|\Lambda(Q_{1},Q_{2})|)j^{\prime}+|\Lambda(Q_{1},Q_{2})|\left(\frac{k}{2}-j^{\prime}\right),$
$\displaystyle
j_{12}^{\prime}=(1-|\Lambda(Q_{1},Q_{2})|)j_{12}+|\Lambda(Q_{1},Q_{2})|\left(\frac{k}{2}-j_{12}\right),$
$\displaystyle
j_{23}^{\prime}=(1-|\Lambda(Q_{2},Q_{3})|)j_{23}+|\Lambda(Q_{2},Q_{3})|\left(\frac{k}{2}-j_{23}\right).$
(59)

The $R$-symbol of the MTC $\mathcal{D}$ is given by

$\displaystyle
R^{[(j_{1},Q_{1})],[(j_{2},Q_{2})]}_{[(j,Q)]}=R^{j_{1},j_{2}}_{j^{\prime}}e^{-\frac{\pi\mathrm{i}}{\sigma_{H}}Q_{1}Q_{2}}$
(60)

where
$j^{\prime}=(1-|\Lambda(Q_{1},Q_{2})|)j+|\Lambda(Q_{1},Q_{2})|\left(\frac{k}{2}-j\right)$.

As an example, we can consider the case of $\mathcal{C}={\rm SU}(2)_{2}$ with
Hall conductance $\sigma_{H}=1$, namely $k=2$ and $\sigma_{H}=1$. After
gauging the U(1) symmetry, the resulting MTC $\mathcal{D}$ has three anyons
$[(0,0)]$, $[(\frac{1}{2},\frac{1}{2})]$ and $[(1,0)]$. The fusion rule given
in Eq. (55) matches that of the Ising MTC. One can further check that the
$F$\- and $R$-symbols of $\mathcal{D}$ obtained from Eq. (58) and Eq. (60)
match those of the Ising MTC (up to a gauge transformation).

If we instead consider the case of SU$(2)_{2}$ with the Hall conductance
$\sigma_{H}=-1$, namely $k=2$ and $\sigma_{H}=-1$. The resulting MTC obtained
from gauging the U(1) symmetry becomes the Spin(5)1 MTC.

##  Appendix C Gauge fixing for the group of transparent Abelian anyons

Here, we focus on the $F$\- and $R$-symbols of $\mathcal{C}^{\prime}$
restricted to the group of transparent Abelian anyons
$\mathcal{T}=\\{\tau^{k}\\}_{k\in\mathbb{Z}}$. Notice that, in the cases we
are interested in, $\mathcal{T}$ has a fusion rule isomorphic to the Abelian
group $\mathbb{Z}$. The $F$-symbol restricted to $\mathcal{T}$ depends only on
the three superscripts and, hence, can be written as
$F^{\tau^{k},\tau^{l},\tau^{m}}$. The remaining (and suppressed) indices can
be inferred from the three superscripts. With the anyon fusion in
$\mathcal{T}$ viewed as the group multiplication in $\mathbb{Z}$, we can
identify the $F$-symbol restricted to $\mathcal{T}$ as an element in the group
cohomology $\mathcal{H}^{3}[\mathbb{Z},\mathrm{U}(1)]$, which turns out to be
trivial. i.e. $\mathcal{H}^{3}[\mathbb{Z},\mathrm{U}(1)]=\mathbb{Z}_{1}$.
Therefore, it is always possible to choose a gauge such that the $F$-symbol
restricted to $\mathcal{T}$ is completely trivial.

The $F$\- and $R$-symbols restricted to $\mathcal{T}$ must satisfy the hexagon
equations. When the $F$-symbols restricted to $\mathcal{T}$ are completely
trivial, the hexagon equations read

$R^{\tau^{k+l},\tau^{m}}=R^{\tau^{k},\tau^{m}}R^{\tau^{l},\tau^{m}},~{}~{}~{}~{}R^{\tau^{m},\tau^{k+l}}=R^{\tau^{m},\tau^{k}}R^{\tau^{m},\tau^{l}}.$
(61)

In particular, they imply that $R^{\tau^{k},\tau^{l}}=(R^{\tau,\tau})^{kl}=1$
because the transparent Abelian anyon $\tau$ under our consideration has a
bosonic self-statistics, namely $R^{\tau,\tau}=\theta_{\tau}=1$.

##  Appendix D Chiral central charge after gauging the U(1) symmetry

Denote by $c_{-}^{\prime}$ the chiral central charge of the gauged system. We
focus on the case where the Hall conductance $\sigma_{H}$ is non-zero, i.e.
$\sigma_{H}\neq 0$. We prove below that $c_{-}^{\prime}\equiv
c_{-}-\operatorname{sgn}\sigma_{H}\mod 8$, where $c_{-}$ is the chiral central
charge of the original ungauged system.

In a general MTC $\mathcal{C}$, we have the generalized Gauss-Milgram sum:

$\frac{1}{D_{\mathcal{C}}}\sum_{a}d_{a}^{2}\theta_{a}=e^{\frac{2\pi\mathrm{i}c_{-}}{8}}.$
(62)

where $d_{a}$ is the quantum dimension of the anyon $a\in\mathcal{C}$,
$D_{\mathcal{C}}\equiv\sqrt{\sum_{a\in\mathcal{C}}d_{a}^{2}}$ is the total
quantum dimension of the MTC $\mathcal{C}$ and $c_{-}$ is the chiral central
charge of $\mathcal{C}$.

Note that a similar identity holds for a general premodular category
$\mathcal{K}$ whose subcategory of transparent anyons $\mathcal{A}$ consists
of only Abelian anyons with bosonic self-statistics Ng  _et al._ (2018):

$\frac{1}{\sqrt{|\mathcal{A}|}D_{\mathcal{K}}}\sum_{a\in\mathcal{K}}d_{a}^{2}\theta_{a}=e^{\frac{2\pi\mathrm{i}c_{-}}{8}},$
(63)

where $d_{a}$ is the quantum dimension of the anyon $a\in\mathcal{K}$,
$D_{\mathcal{K}}\equiv\sqrt{\sum_{a\in\mathcal{K}}d_{a}^{2}}$ is the total
quantum dimension of the premodular category $\mathcal{K}$ and $c_{-}$ is the
chiral central charge of the MTC obtained from condensing $\mathcal{A}$ in the
premodular category $\mathcal{K}$. Here, $|\mathcal{A}|$ is the number of
Abelian anyons in the transparent subcategory $\mathcal{A}$ and is assumed to
be finite.

In this work, we start with an U(1)-symmetric MTC $\mathcal{C}$ with a chiral
central charge $c_{-}$ and a Hall conductance $\sigma_{H}$. Assuming
$\sigma_{H}\neq 0$, we want to calculate the chiral central charge
$c_{-}^{\prime}$ of the MTC $\mathcal{D}$ obtained from gauging the U(1)
symmetry of $\mathcal{C}$. According to the general procedure of gauging
presented in App. B, the MTC $\mathcal{D}$ can be viewed as a result of
condensing a group of transparent Abelian bosons in the intermediate
premodular category $\mathcal{D}_{\rm int}$ introduced in App. B.2. The
intermediate category $\mathcal{D}_{\rm int}$ is obtained from the infinite
category $\mathcal{C}^{\prime}$ by condensing $(1,s\sigma_{H})$, as discussed
in App. B.2. In the following, we calculate chiral central charge
$c^{\prime}_{-}$ of the MTC $\mathcal{D}$ using the intermediate premodular
category $\mathcal{D}_{\rm int}$.

Recall that the simple objects in this premodular category $\mathcal{D}_{\rm
int}$ are labeled by $[(a,Q_{a})]$ where $a\in\mathcal{C}$ and the value of
$Q_{a}$ is restricted to the range $0\leq Q_{a}<|s\sigma_{H}|$. Also, we
recall the $F$ -and $R$\- symbols of this category $\mathcal{D}_{\rm int}$ are
given by

$\begin{split}\left(F^{[(a,Q_{a})],[(b,Q_{b})],[(c,Q_{c})]}_{[(d,Q_{d})]}\right)_{[(e,Q_{e})],[(f,Q_{f})]}&=\left(F^{a,b,c}_{d}\right)_{e,f}e^{\frac{\pi\mathrm{i}}{\sigma_{H}}\lceil
Q_{a}\rfloor_{|s\sigma_{H}|}(\lceil Q_{b}+Q_{c}\rfloor_{|s\sigma_{H}|}-\lceil
Q_{b}\rfloor_{|s\sigma_{H}|}-\lceil Q_{c}\rfloor_{|s\sigma_{H}|})},\\\
R^{[(a,Q_{a})],[(b,Q_{b})]}_{[(c,Q_{c})]}&=R^{ab}_{c}e^{-\frac{\pi\mathrm{i}}{\sigma_{H}}\lceil
Q_{a}\rfloor_{|s\sigma_{H}|}\lceil Q_{b}\rfloor_{|s\sigma_{H}|}}.\end{split}$
(64)

where $\left(F^{a,b,c}_{d}\right)_{e,f}$ and $R^{ab}_{c}$ are the $F$\- and
$R$-symbols of the original MTC $\mathcal{C}$ before gauging the U(1)
symmetry. The total quantum dimension $D_{\mathcal{D}_{\rm int}}$ of this
premodular category $\mathcal{D}_{\rm int}$ is given by $D_{\mathcal{D}_{\rm
int}}=\sqrt{|s\sigma_{H}|}D_{\mathcal{C}}$. In $\mathcal{D}_{\rm int}$, The
quantum dimension $d_{[(a,Q_{a})]}$ and the topological twist factor
$\theta_{[(a,Q_{a})]}$ are given by

$\displaystyle
d_{[(a,Q_{a})]}=d_{a},~{}~{}~{}~{}\theta_{[(a,Q_{a})]}=\theta_{a}e^{-\frac{\pi\mathrm{i}}{\sigma_{H}}\lceil
Q_{a}\rfloor_{|s\sigma_{H}|}^{2}},$ (65)

where $d_{a}$ and $\theta_{a}$ are the quantum dimension and the topological
twist factor of the anyon $a$ in the original MTC $\mathcal{C}$. To obtain the
MTC $\mathcal{D}$ from premodular category $\mathcal{D}_{\rm int}$, one needs
to condense the group of transparent Abelian bosons
$\mathcal{A}\subset\mathcal{D}_{\rm int}$ generated by $[(v,\sigma_{H})]$.
Notice that $|\mathcal{A}|=s$.

Before performing the generalized Gauss-Milgram sum, it is useful to notice
that, for a given $a\in\mathcal{C}$, the allowed anyons
$[(a,Q_{a})]\in\mathcal{D}_{\rm int}$ have
$Q_{a}\in\\{q_{a},q_{a}+1,...,q_{a}+|s\sigma_{H}|-1\\}$, where $q_{a}$ is
defined via $M_{av}=e^{2\pi\mathrm{i}q_{a}}$ (using the braiding $M_{av}$ of
the original MTC $\mathcal{C}$) and $0\leq q_{a}<1$. Therefore, we can write
the following generalized Gauss-Milgram sum

$\begin{split}e^{\frac{2\pi\mathrm{i}c_{-}^{\prime}}{8}}&=\frac{1}{\sqrt{|\mathcal{A}|}D_{\mathcal{D}_{\rm
int}}}\sum_{[(a,Q_{a})]\in D_{\mathcal{D}_{\rm
int}}}d_{[(a,Q_{a})]}^{2}\theta_{[(a,Q_{a})]}\\\
&=\frac{1}{s\sqrt{|\sigma_{H}|}D_{\mathcal{C}}}\sum_{a\in\mathcal{C}}\sum_{k=0}^{s\sigma_{H}-1}d_{a}^{2}\theta_{a}e^{-\frac{\pi\mathrm{i}}{\sigma_{H}}(q_{a}+k)^{2}}.\end{split}$
(66)

First we perform the summation over $k$:

$\displaystyle\begin{split}\sum_{k=0}^{s\sigma_{H}-1}e^{-\frac{\mathrm{i}\pi}{\sigma_{H}}(q_{a}+k)^{2}}&=\frac{1}{s}e^{-\frac{\pi\mathrm{i}}{\sigma_{H}}q_{a}^{2}}\sum_{k=0}^{s^{2}\sigma_{H}-1}e^{-\frac{\pi\mathrm{i}}{s^{2}\sigma_{H}}(s^{2}k^{2}+2q_{a}s^{2}k)}\\\
&=\frac{1}{s}\sqrt{|\sigma_{H}|}e^{-\frac{\pi\mathrm{i}}{4}\operatorname{sgn}\sigma_{H}}\sum_{k=0}^{s^{2}-1}e^{\frac{\pi\mathrm{i}}{s^{2}}(s^{2}\sigma_{H}k^{2}+2q_{a}s^{2}k)}\\\
&=\frac{1}{s}\sqrt{|\sigma_{H}|}e^{-\frac{\pi\mathrm{i}}{4}\operatorname{sgn}\sigma_{H}}\sum_{k=0}^{s^{2}-1}e^{\pi\mathrm{i}(\sigma_{H}k^{2}+2q_{a}k)}\\\
&=\sqrt{|\sigma_{H}|}e^{-\frac{\pi\mathrm{i}}{4}\operatorname{sgn}\sigma_{H}}\sum_{k=0}^{s-1}e^{\pi\mathrm{i}(\sigma_{H}k^{2}+2q_{a}k)}\\\
&=\sqrt{|\sigma_{H}|}e^{\frac{-\pi\mathrm{i}}{4}\operatorname{sgn}\sigma_{H}}\sum_{k=0}^{s-1}\theta_{v^{k}}M_{v^{k},a}.\end{split}$
(67)

For the second equality we apply the quadratic reciprocal law for Gauss sums.
Now, the full generalized Gauss-Milgram sum of $\mathcal{D}_{\rm int}$ can be
evaluated:

$\begin{split}e^{\frac{2\pi\mathrm{i}c_{-}^{\prime}}{8}}&=\frac{1}{sD_{\mathcal{C}}}e^{-\frac{\pi\mathrm{i}}{4}\operatorname{sgn}\sigma_{H}}\sum_{a\in\mathcal{C}}\sum_{k=0}^{s-1}d_{a}^{2}\theta_{a}\theta_{v^{k}}M_{v^{k},a}\\\
&=\frac{1}{sD_{\mathcal{C}}}e^{-\frac{\pi\mathrm{i}}{4}\operatorname{sgn}\sigma_{H}}\sum_{k=0}^{s-1}\sum_{a\in\mathcal{C}}d_{a\times
v^{k}}^{2}\theta_{a\times v^{k}}\\\
&=e^{\frac{\pi\mathrm{i}}{4}(c_{-}-\operatorname{sgn}\sigma_{H})}\end{split}.$
(68)

Hence, we find $c_{-}^{\prime}=c_{-}-\operatorname{sgn}\sigma_{H}$ mod 8.

##  Appendix E General ${\rm U}(N)_{k,k+Nk^{\prime}}$ MTC from gauging the
${\rm U}(1)$ symmetry in ${\rm SU}(N)_{k}$

The ${\rm U}(N)_{k,k+Nk^{\prime}}$ MTC with $k,k^{\prime}\in\mathbb{Z}$ and
with $k+k^{\prime}$ an even integer describes the bosonic topological order
associated with the 2+1d Chern-Simons theory

$\displaystyle\mathcal{L}=-\frac{k}{4\pi}\operatorname{Tr}\left(bdb-\frac{2\mathrm{i}}{3}b^{3}\right)-\frac{k^{\prime}}{4\pi}(\operatorname{Tr}b)d(\operatorname{Tr}b),$
(69)

where $b$ is the U$(N)$ gauge connection which is a $N\times N$-matrix-valued
1-form. Let’s denote the traceless part of the gauge connection $b$ as the
1-form gauge field $a$ and the trace as a gauge field $A$, i.e.
$\operatorname{Tr}b=A$. The gauge field $a$ can be interpreted as an SU$(N)$
gauge connection and the gauge field $A$ as a U(1) gauge connection. When we
treat the gauge field $a$ as a dynamical field while the gauge field $A$ only
as a static background gauge field, the Lagrangian in Eq. (69) describes a
$\mathcal{C}={\rm SU}(N)_{k}$ topological order with a U(1) 0-form global
symmetry. The associated Hall conductance is given by
$\sigma_{H}=-\frac{k}{N}-k^{\prime}$. The $2\pi$ flux of the U(1) symmetry is
naturally associated with the Abelian anyon $v\in{\rm SU}(N)_{k}$ whose Wilson
line generates the $\mathbb{Z}_{N}$ 1-form symmetry of the ${\rm SU}(N)_{k}$
topological order. This Abelian anyon $v$ has a $\mathbb{Z}_{N}$ fusion rule
(in the ${\rm SU}(N)_{k}$ MTC) and a topological spin
$\theta_{v}=e^{\mathrm{i}2\pi\frac{(N-1)k}{2N}}$. The aforementioned
requirement that $k+k^{\prime}$ is even ensures the consistency condition
$e^{i\pi\sigma_{H}}=\theta_{v}$. Obviously, when we gauge this U(1) 0-form
global symmetry, we restore the dynamics of the gauge field
$A=\operatorname{Tr}b$. Hence, the so-obtained topological order after gauging
is the ${\rm U}(N)_{k,k+Nk^{\prime}}$ topological order that is described by
the Lagrangian Eq. (69) with a fully dynamical the U$(N)$ gauge field $b$.

The ${\rm U}(N)_{k,k+Nk^{\prime}}$ topological order can also be written as
${\rm U}(N)_{k,k+Nk^{\prime}}=\frac{{\rm SU}(N)_{k}\times{\rm
U}(1)_{N(k+Nk^{\prime})}}{\mathbb{Z}_{N}}$Hsin and Seiberg (2016). Hence, it
can be constructed from an anyon condensation in $\mathcal{B}_{0}={\rm
SU}(N)_{k}\times{\rm U}(1)_{N(k+Nk^{\prime})}$. The anyon to be condensed here
is the composite of $v\in{\rm SU}(N)_{k}$ and the Abelian anyon
${x}^{-(k+Nk^{\prime})}\in{\rm U}(1)_{N(k+Nk^{\prime})}$ where $x\in{\rm
U}(1)_{N(k+Nk^{\prime})}$ here denotes the anyon that generates the entire
$\mathbb{Z}_{N(k+Nk^{\prime})}$ fusion algebra of the ${\rm
U}(1)_{N(k+Nk^{\prime})}$ sector. The Abelian anyon
${x}^{-(k+Nk^{\prime})}\in{\rm U}(1)_{N(k+Nk^{\prime})}$ also has a
$\mathbb{Z}_{N}$ fusion rule (in the ${\rm U}(1)_{N(k+Nk^{\prime})}$ MTC) and
a topological spin
$\theta_{x^{-(k+Nk^{\prime})}}=e^{\frac{\mathrm{i}2\pi(k+Nk^{\prime})}{2N}}$.
The composite Abelian anyon $(v,x^{-(k+Nk^{\prime})})\in\mathcal{B}_{0}={\rm
SU}(N)_{k}\times{\rm U}(1)_{N(k+Nk^{\prime})}$ has a bosonic self-statistics
and, hence, is allowed to condense.

Similar to the discussion in Sec. III.2, we first consider the premodular sub-
category $\mathcal{B}_{1}\subset\mathcal{B}_{0}={\rm SU}(N)_{k}\times{\rm
U}(1)_{N(k+Nk^{\prime})}$ that braids trivially with
$(v,x^{-(k+Nk^{\prime})})$. One can show that this sub-category is completely
identical to the intermediate premodular category $\mathcal{D}_{\rm int}$
obtained from applying the general procedures for gauging the U(1) symmetry
described in App. B.2 to the SU$(N)_{k}$ MTC with a Hall conductance of
$\sigma_{H}=-\frac{k}{N}-k^{\prime}$. Here, $s=N$ because the vison $v\in{\rm
SU}(N)_{k}$ associated with the $2\pi$ flux has a $\mathbb{Z}_{N}$ fusion
rule. Further condensation of $(v,\sigma_{H})\in\mathcal{D}_{\rm int}$ (which
is needed for completing the full U(1) gauging procedure described in App. B)
in the intermediate category $\mathcal{D}_{\rm int}$ is equivalent to the
condensation of the Abelian anyon $(v,x^{-(k+Nk^{\prime})})$ in
$\mathcal{B}_{1}$. This condensation yields the ${\rm U}(N)_{k,k+Nk^{\prime}}$
MTC as the final result. The discussion of the $\mathbb{Z}_{k}$ parafermion in
Sec. III.2 is a special case of the general discussion here.

Also, we notice that when the Hall conductance vanishes, i.e. when
$\sigma_{H}=-\frac{k}{N}-k^{\prime}=0$, the field theory Eq. (69) indicates
that, after the U(1) symmetry is gauged, the resulting theory is given by
${\rm U}(N)_{k,0}=\frac{{\rm SU}(N)_{k}}{\mathbb{Z}_{N}}$. The theory
$\frac{{\rm SU}(N)_{k}}{\mathbb{Z}_{N}}$ is exactly given by condensing the
vison $v$, which generates the $\mathbb{Z}_{N}$ 1-form symmetry of ${\rm
SU}(N)_{k}$, in the ungauged theory ${\rm SU}(N)_{k}$. Remember that $k$ and
$k^{\prime}$ are both integers and $k+k^{\prime}$ is required to be even. A
vanishing Hall conductance only occurs when (1) $k$ is an even-integer
multiple of $N$ for even $N$ or (2) $k$ is an integer multiple of $N$ for odd
$N$. In these scenarios, the vision $v$ has a bosonic self-statistics
$\theta_{v}=1$ and is allowed to condense.

##  Appendix F Holographic viewpoint of gauging the U(1) symmetry

For a (2+1)d topological order $\mathcal{C}$ with a global (0-form) symmetry
$G$, one way to think about the coupling between the (2+1)d topological order
to either the background $G$ gauge field or a dynamical $G$-gauge field (after
gauging the global symmetry $G$) is via a “holographic” viewpoint in which the
(2+1)d spacetime where the (2+1)d topological order resides is the boundary of
a (3+1)d spacetime manifold where the $G$ gauge field extends. This viewpoint
provides a useful tool in characterizing the possible ’t Hooft anomalies of
(2+1)d topological order under the a global symmetry $G$ (see Ref. Chen  _et
al._ , 2015; Metlitski  _et al._ , 2014; Bonderson  _et al._ , 2013 for early
examples) in which context the $G$ gauge field is often treated as a static
background gauge field. In this appendix, we will discuss such a similar
holographic viewpoint for gauging a 0-form global $G$ symmetry of a (2+1)d
topological order. In particular, we are interested in the case where
$G=\mathrm{U}(1)$.

As a start, let’s first discus the case when the 0-form global symmetry group
$G$ is a finite group. For simplicity, let’s consider the $2+1$d topological
order $\mathcal{C}$ living on the 2-dimensional spatial $x$-$y$ plane at $z=0$
while the 3-dimensional bulk is the entire “half-space” with $z\leq 0$. The
1-form $G$ gauge field, background or dynamical, lives in the the entire bulk
at $z\leq 0$ and couples to the 2+1d topological order $\mathcal{C}$ on the
boundary at $z=0$. Assuming the global symmetry $G$ of the (2+1)d topological
order $\mathcal{C}$ is free of anomaly, we can take (3+1)d bulk to be in a
trivial $G$-SPT phase before gauging the global symmetry $G$. After we gauge
the symmetry $G$, the bulk hosts a un-twisted dynamical (and deconfined)
1-form $G$ gauge theory, which itself is a (3+1)d topological order with a
finite energy gap. What is the (2+1)d topological order $\mathcal{D}$ that is
the outcome of gauging the $G$ symmetry of $\mathcal{C}$ purely within (2+1)d?
To answer this question in the holographic viewpoint, we can consider the bulk
to resides on a “slab of finite width”, say the 3d space with $-W\leq z\leq
0$. Here, the $x$-$y$ plane at the boundary at $z=0$ is still where the
original topological order $\mathcal{C}$ resides. On the boundary at $z=-W$,
we impose the gapped boundary condition for the (3+1)d dynamical 1-form $G$
gauge theory such that any $G$-flux is allowed to terminate (without energy
cost) at the boundary at $z=-W$. Since the entire bulk has a finite width $W$,
the total system, including the bulk and the boundaries at $z=0$ and at
$z=-W$, can be viewed as an effective (2+1)d system. This slab construction
maintain a finite energy gap at all times and, therefore, yields a (2+1)d
topological order that should be identified as $\mathcal{D}$.

Now, let’s consider the case of a (2+1)d topological order $\mathcal{C}$ with
a global 0-form symmetry $G=\mathrm{U}(1)$. We can still consider the (3+1)d
bulk defined on the 3D space with $z\leq 0$ and the (2+1)d topological order
$\mathcal{C}$ living on the boundary at $z=0$. Before we gauge the U(1)
symmetry, we still require the background ground U(1) 1-form gauge field to be
defined in the entire bulk at $z\leq 0$. This background U(1) gauge field
couples to the (2+1)d topological order $\mathcal{C}$ at the boundary at
$z=0$. Since the U(1) symmetry of the topological order $\mathcal{C}$ is free
of any ’t Hooft anomaly, the (3+1)d bulk is in the trivial (3+1)d U(1)-SPT
phase. In fact, there is no non-trivial SPT phase with a single 0-form global
U(1) symmetry in (3+1)d. Now, let’s gauge the U(1) symmetry and, thereby,
promoting the background U(1) 1-form gauge field to a dynamical one. We can
start by first letting the dynamics of the bulk U(1) gauge field to be
governed by the Maxwell theory. Such a bulk theory is obviously gapless. In
the bulk, any integer U(1) charge is allowed, while on the boundary at $z=0$
the infinite set of allowed combinations of anyons in $\mathcal{C}$ and U(1)
charges are captured by the infinite category $\mathcal{C}^{\prime}$ defined
via Eq. (18) and 21 assuming $\sigma_{H}\neq 0$. Directly applying the slab
construction to such a bulk theory does not lead any regular (2+1)d
topological order (that should have an energy gap and finitely many anyons).
We can nevertheless continue to analyze this system.

There is a special excitation on the boundary given by the anyon
$(v,\sigma_{H})\in\mathcal{C}^{\prime}$ where $v\in\mathcal{C}$, the vison, is
associated with the $2\pi$ flux of the U(1) gauge field. The U(1) charge
$\sigma_{H}$ carried by this anyon $(v,\sigma_{H})\in\mathcal{C}^{\prime}$ is
equal to the Hall conductance of the topological order $\mathcal{C}$ on the
boundary. When a monopole of the U(1) gauge field, which carries a total of
$2\pi$ flux, tunnels through boundary at $z=0$ into the bulk, it leaves behind
on the boundary the composite of the anyon $v\in\mathcal{C}$ and the U(1)
charge $\sigma_{H}$ due to the Hall effect generated by the topological order
$\mathcal{C}$ on the boundary. Therefore, the anyon
$(v,\sigma_{H})\in\mathcal{C}^{\prime}$ is created when a U(1) monopole
tunnels into the bulk. This analysis also suggests that the bulk Maxwell
theory has a non-trivial $\theta$-term $\int\frac{\sigma_{H}}{4\pi}dAdA$
(where $A$ is the 1-form U(1) gauge field).

Now, we condense the U(1) monopoles in the bulk to drive the bulk into a
confined phase of the U(1) gauge field. The confined phase is gapped and free
of any (3+1)d topological order. The condensation of the U(1) monopoles in the
bulk leads to the condensation of the anyon
$(v,\sigma_{H})\in\mathcal{C}^{\prime}$ on the boundary. Therefore, the
condensation of the anyon $(v,\sigma_{H})\in\mathcal{C}^{\prime}$ on the
boundary yields the topological order $\mathcal{D}$ that is the outcome of
gauging the U(1) symmetry of the original topological order $\mathcal{C}$.
Here, it may seem that, in the holographic viewpoint, the confinement of the
bulk U(1) gauge field via monopole condensation is an extra step beyond simply
gauging the U(1) symmetry. We argue that this is natural and necessary because
a simple U(1) Maxwell gauge theory in purely (2+1)d does confine
automatically. Technically speaking, one should again consider monopole
condensation in the bulk that is a slab of finite width, say the 3d space with
$-W\leq z\leq 0$. In this case, we can choose the boundary condition at $z=-W$
such that the monopole condensation simply leads to the confinement of the
U(1) gauge field both in the bulk and at the boundary $z=-W$. The non-trivial
degrees of freedom left all resides on the boundary at $z=0$ and are captured
by the (2+1)d topological order $\mathcal{D}$ (which is obtained from
condensing $(v,\sigma_{H})$ in the infinite category $\mathcal{C}^{\prime}$).

So far, we have assumed that $\sigma_{H}\neq 0$. In fact, one can consider the
exact same setup when $\sigma_{H}=0$. In this case, when the bulk is governed
by the gapless 3+1d Maxwell theory, there are still an infinite set of
excitations given by the allowed combinations of anyons in $\mathcal{C}$ and
U(1) charges. When the monopoles of $A$ condense driving the 3+1d bulk into
the confined phase, the vison $v$, which now carries a vanishing U(1) charge,
also condenses on the 2+1d boundary. The condensation of the vison $v$ drives
the original MTC $\mathcal{C}$ into a new MTC $\mathcal{D}$ that is equivalent
to the result of gauging the U(1) symmetry of the original MTC $\mathcal{C}$.

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