where we’ve used the condition that ${\rm U}(N)_{k_{1},k_{2}}$ is a properly
quantized bosonic Chern-Simons term for a ${\rm U}(N)$ gauge group, and
$nr^{\prime}$ is even in bosonic theory ${\mathcal{C}}$.

$\mathcal{X}$ has an Abelian anyon $\rho$ that satisfies
$\rho^{r^{\prime}}=1$. It has topological spin

$\tilde{\theta}_{\rho}=\exp\left\\{\mathrm{i}2\pi\left(\frac{k_{1}(N-1)}{2N}+\frac{k_{2}-nN/r^{\prime}}{2N}\right)\right\\}=\exp\left\\{\mathrm{i}2\pi\left(\left(\frac{k_{1}}{2}+\frac{k_{2}-k_{1}}{2N}\right)-\frac{n}{2r^{\prime}}\right)\right\\}=\exp\left\\{-\mathrm{i}2\pi\frac{n}{2r^{\prime}}\right\\},$
(98)

where we’ve again used the condition that ${\rm U}(N)_{k_{1},k_{2}}$ is a
properly quantized bosonic Chern-Simons term for a ${\rm U}(N)$ gauge field.

Thus, the theory $\mathcal{D}$ after gauging the 0-form symmetry $G={\rm
U}(N)$ is

$\mathcal{D}={{\mathcal{C}}\boxtimes{\mathcal{X}}\over\mathbb{Z}_{r^{\prime}}}={{\mathcal{C}}\boxtimes{\rm
SU}(N)_{k_{1}}\boxtimes\tilde{\rm U}(1)_{\frac{r^{\prime
2}}{\gcd(N,r^{\prime})^{2}}N(k_{2}-nN/r^{\prime})}\over\mathbb{Z}_{Nr^{\prime}/\gcd(N,r^{\prime})}}~{},$
(99)

where the quotient is generated by condensing $v\otimes\rho$, whose worldline
generates the $\mathbb{Z}_{r^{\prime}}$ one-form symmetry in
${\mathcal{C}}\boxtimes{\mathcal{X}}$ and the
$\mathbb{Z}_{Nr^{\prime}/\gcd(N,r^{\prime})}$ one-form symmetry in
${\mathcal{C}}\boxtimes{\rm SU}(N)_{k_{1}}\boxtimes\tilde{\rm
U}(1)_{\frac{r^{\prime 2}}{\gcd(N,r^{\prime})^{2}}N(k_{2}-nN/r^{\prime})}$.
This result of $\mathcal{D}$ is equivalent to the result Eq. (50) in Sec.
III.3. We simply need to identify the Abelian anyon $v_{1}$ in this subsection
with the Abelian anyon $v_{1}$ introduced in Sec. III.3 and set
$r=\frac{r^{\prime}}{\gcd(r^{\prime},N)}$. Note that for the Abelian anyon
$v_{1}$ that characterizes the symmetry fractionalization pattern in
$\mathcal{C}$, when $v_{1}$ has a $\mathbb{Z}_{r^{\prime}}$-fusion rule, the
minimal choice of $K$ in the group extension Eq. (47) is given by
$K=\mathbb{Z}_{rN}=\mathbb{Z}_{Nr^{\prime}/\gcd(r^{\prime},N)}$. The Abelian
anyon $\rho\in{\rm SU}(N)_{k_{1}}\boxtimes\tilde{\rm U}(1)_{\frac{r^{\prime
2}}{\gcd(N,r^{\prime})^{2}}N(k_{2}-nN/r^{\prime})}$ appearing in this
subsection should be identified with the Abelian anyon $(s,s^{\prime})$
introduced in Sec. III.3.

###  IV.4 Example: $G={\rm U}(N)/\mathbb{Z}_{M}$

####  IV.4.1 Symmetry fractionalization

Let’s consider a 2+1d topological phase $\mathcal{C}$ with a 0-form $G={\rm
U}(N)/\mathbb{Z}_{M}$ global symmetry. The ${\rm U}(N)/\mathbb{Z}_{M}$
symmetry fractionalization in $\mathcal{C}$ can be described by coupling
$\mathcal{C}$ to the ${\rm U}(N)/\mathbb{Z}_{M}$ background gauge field $A$
via the $\mathcal{A}$-valued two-form gauge field $B$:

$B=\mu[A]~{},$ (100)

where $\mu\in H^{2}({\rm B}{\rm U}(N)/\mathbb{Z}_{M},\mathcal{A})$.

We will discuss $\mu[A]$ in more details as follows. As discussed in Sec.
III.4.1, for a ${\rm U}(N)/\mathbb{Z}_{M}$ background gauge field $A$, or
equivalently for a ${\rm U}(N)/\mathbb{Z}_{M}$ gauge bundle, there are three
type of characteristic classes: $w^{(M)}[A]$, $w^{(N)}[A]$ and $c_{1}$. We
will drop the “$[A]$” part to simplify the notation in the following. Recall
that $w^{(M)}[A]$ is a $\mathbb{Z}_{M}$-valued two-cocycle, $w^{(N)}[A]$ is
$\mathbb{Z}_{N}$-valued two-cocycle and $c_{1}$ is a $\mathbb{Z}$-valued two-
cocycle, which satisfy the relation Eq. (53):

$c_{1}=\frac{M}{L}w_{2}^{(N)}+\frac{N}{L}w_{2}^{(M)}\quad\text{mod
}\frac{NM}{L}$ (101)

with $L\equiv\gcd(M,N)$. The fractionalization class $\mu\in H^{2}({\rm B}{\rm
U}(N)/\mathbb{Z}_{M},{\mathcal{A}})$ can be expressed as a linear combination
of $w_{2}^{(N)},w_{2}^{(M)},c_{1}$

$\mu[A]=\iota^{(N)}w_{2}^{(N)}+\iota^{(M)}w_{2}^{(M)}+\iota c_{1}~{},$ (102)

with homomorphisms $\iota^{(N)}:\mathbb{Z}_{N}\rightarrow{\mathcal{A}}$,
$\iota^{(M)}:\mathbb{Z}_{M}\rightarrow{\mathcal{A}}$, and
$\iota:\mathbb{Z}\rightarrow{\mathcal{A}}$. More explicitly, in terms of
${\mathcal{A}}=\prod_{i}\mathbb{Z}_{N_{i}}$, the expression of $\mu[A]$ is

$\mu[A]_{i}=\alpha_{i}w_{2}^{(N)}+\beta_{i}w_{2}^{(M)}+\gamma_{i}c_{1}\text{
mod }N_{i},\quad\alpha_{i},\beta_{i},\gamma_{i}\in\mathbb{Z}_{N_{i}}~{},$
(103)

where $\mu[A]_{i}$ is the restriction of the $\mathcal{A}$-valued two-cocycle
to the $\mathbb{Z}_{N_{i}}$ factor of $\mathcal{A}$.

We remark that one might worry that the coefficients $\alpha_{i}$,
$\beta_{i}$, $\gamma_{i}$ could be fractional due to the relation Eq. 101.
Given ${\mathcal{A}}=\prod\mathbb{Z}_{N_{i}}$, such fractional terms can be
written as

$\frac{p_{i}}{d_{i}}\left(c_{1}-\frac{M}{L}w_{2}^{(N)}-\frac{N}{L}w_{2}^{(M)}\right)\text{
mod }N_{i}~{},$ (104)

where $d_{i},p_{i}$ are integers, and $d_{i}$ is a divisor of $MN/L$, such
that the above expression is an integer, but the coefficients for
$c_{1},w_{2}^{(N)},w_{2}^{(M)}$ separately may not be integers. For such term
to be well-defined, in particular invariant under shifting $w_{2}^{(N)}$ by a
multiple of $N$, we need $p_{i}MN/(Ld_{i})=0$ mod $N_{i}$. But this is
equivalent to setting the above expression to be zero modulo $N_{i}$, since
$c_{1}-\frac{M}{L}w_{2}^{(N)}-\frac{N}{L}w_{2}^{(M)}$ is a multiple of $MN/L$
by the relation (101). Another way to see this is that we can first express
$\mu[A]$ in terms of independent $w_{2}^{(N)},w_{2}^{(M)},c_{1}$, then impose
the relation (101) by a Lagrangian multiplier. Then it is clear that the
coefficients $\alpha_{i}$, $\beta_{i}$, $\gamma_{i}$ represent homomorphisms
from $\mathbb{Z}_{N}\times\mathbb{Z}_{M}\times\mathbb{Z}$ to ${\mathcal{A}}$.

The image of the generator of $\mathbb{Z}_{N}$ under the homomorphism
$\iota^{(N)}$ is an Abelian anyon
$u_{1}=\\{\alpha_{i}\\}\in\mathcal{A}=\prod_{i}\mathbb{Z}_{N_{i}}$. Similarly,
the images of the generators of $\mathbb{Z}_{M}$ and $\mathbb{Z}$ under
$\iota^{(M)}$ and $\iota$ respectively are Abelian anyons
$u_{2}=\\{\beta_{i}\\}$ and $u_{3}=\\{\gamma_{i}\\}$. Let’s denote the order
of $u_{3}$ by $r^{\prime}$. Also considering that $\mu[A]$ must be invariant
under changing $w_{2}^{(N)},w_{2}^{(M)}$ by multiples of $N$ and $M$
respectively, the Abelian anyons $u_{1}$, $u_{2}$ and $u_{3}$ satisfy

$u_{1}^{N}=1,\quad u_{2}^{M}=1,\quad u_{3}^{r^{\prime}}=1.$ (105)

We can view the three Abelian anyons $(u_{1},u_{2},u_{3})$ as a
parameterization of the symmetry fractionalization pattern $\mu\in H^{2}({\rm
B}{\rm U}(N)/\mathbb{Z}_{M},\mathcal{A})$. Since
$w_{2}^{(N)},w_{2}^{(M)},c_{1}$ obey the relation (101), the parameterization
of $\mu$ in terms of $u_{1},u_{2},u_{3}$ has a redundancy given by adding
multiples of $c_{1}-\frac{M}{L}w_{2}^{(N)}-\frac{N}{L}w_{2}^{(M)}$. That is to
say that there are equivalent classes of such parameterizations given by the
equivalence relation:

$(u_{1},u_{2},u_{3})\sim(u_{1}u^{M/L},u_{2}u^{N/L},u_{3}u^{-1}),$ (106)

for any Abelian anyon $u$ that satisfies $u^{MN/L}=1$. Note that this relation
does not mean the anyons are identified in the TQFT. Rather, it means that two
sets of $(u_{1},u_{2},u_{3})$ related in the way shown above correspond to the
same symmetry fractionalization pattern $\mu$. From $(u_{1},u_{2},u_{3})$, we
can define two combinations of Abelian anyons that do not depend on the choice
of $u$:

$v_{1}:=u_{2}u_{3}^{N/L},~{}~{}~{}~{}v_{2}:=u_{1}u_{3}^{M/L},$ (107)

which satisfy

$v_{1}^{Mr^{\prime}/\gcd(r^{\prime},MN/L)}=1,\quad
v_{2}^{Nr^{\prime}/\gcd(r^{\prime},MN/L)}=1,\quad v_{2}^{-N}v_{1}^{M}=1~{}.$
(108)

Since $v_{1}$ and $v_{2}$ are the same within a given equivalent class of
$(u_{1},u_{2},u_{3})$, one can view them as a non-redundant parameterization
of the symmetry fractionalization pattern $\mu$, as we did in Sec. III.4.2. To
more explicitly connect the current discussion with the previous discussion in
Sec. III.4.2, we should identify the parameter $r$ introduced in Sec. III.4.2
with $r^{\prime}/\gcd(r^{\prime},MN/L)$ (or a multiple of
$r^{\prime}/\gcd(r^{\prime},MN/L)$). Then, the Abelian anyons $v_{1},v_{2}$
obey the fusion algebra
$K=(\mathbb{Z}_{Mr}\times\mathbb{Z}_{Nr})/\mathbb{Z}_{r}$ given in Eq. (55).
888 Or equivalently, for $v_{1},v_{2}$ to have all the possible spins as the
anyons that obey $K=(\mathbb{Z}_{Mr}\times\mathbb{Z}_{Nr})/\mathbb{Z}_{r}$
fusion rule, we can consider the case with $r^{\prime}=MNr/L$.

####  IV.4.2 ’t Hooft Anomaly

The anomaly for the $G={\rm U}(N)/\mathbb{Z}_{M}$ symmetry for the
fractionalization class $\mu$ can be computed from the anomaly of the one-form
symmetry via the relation (100). The anomaly for the one-form symmetry is
described by the topological spins and mutual statistics of the Abelian anyons
$u_{1}$, $u_{2}$ and $u_{3}$ in the TQFT ${\mathcal{C}}$, denoted by the
quadratic function $h:{\mathcal{A}}\rightarrow\mathbb{R}/\mathbb{Z}$. The
topological spins and mutual statistics of $u_{1},u_{2},u_{3}$ can be
parameterized as

$\displaystyle\theta_{u_{1}}=e^{2\pi\mathrm{i}n_{1}^{\prime}/(2N)},\quad\theta_{u_{2}}=e^{2\pi\mathrm{i}n_{2}^{\prime}/(2M)},\quad\theta_{u_{3}}=e^{2\pi\mathrm{i}n_{3}^{\prime}/(2r^{\prime})}$
(109) $\displaystyle
M_{u_{1},u_{2}}=e^{2\pi\mathrm{i}n_{12}^{\prime}/\gcd(M,N)},\quad
M_{u_{1},u_{3}}=e^{2\pi\mathrm{i}n_{13}^{\prime}/\gcd(N,r^{\prime})},\quad
M_{u_{2},u_{3}}=e^{2\pi\mathrm{i}n_{23}^{\prime}/\gcd(M,r^{\prime})}~{},$
(110)

for some integers $n_{1}^{\prime}$, $n_{2}^{\prime}$, $n_{3}^{\prime}$,
$n_{12}^{\prime}$, $n_{13}^{\prime}$, and $n_{23}^{\prime}$. Also,
$n_{1}^{\prime}N$, $n_{2}^{\prime}M$ and $n_{3}^{\prime}r^{\prime}$ are all
even integers for the TQFT $\mathcal{C}$ in a bosonic system. We will see in
the following that the ‘t Hooft anomaly only depends on the equivalent classes
of $(u_{1},u_{2},u_{3})$ (defined by the equivalence relation Eq. (106)) and,
hence, is intrinsic to the symmetry fractionalization pattern $\mu$.

Denote by $h$ the $\mathbb{R}/\mathbb{Z}$-valued quadratic function on
${\mathcal{A}}$ that encodes the topological spins of the Abelian anyons in
${\mathcal{A}}$ via $e^{\mathrm{i}2\pi h}$. The anomaly contribution induced
by the one-form symmetry is described by the 3+1d effective action

$\displaystyle S_{\text{anom}}$ $\displaystyle=2\pi\int h[B]=2\pi\int
h[\mu[A]]$ (111)
$\displaystyle=2\pi\int\left(\frac{n_{1}^{\prime}}{2N}{\mathcal{P}}(w_{2}^{(N)})+\frac{n_{2}^{\prime}}{2M}{\mathcal{P}}(w_{2}^{(M)})+\frac{n_{3}^{\prime}}{2r^{\prime}}c_{1}^{2}\right)$
(112)
$\displaystyle\quad+2\pi\int\left(\frac{n_{12}^{\prime}}{L}w_{2}^{(N)}\cup
w_{2}^{(M)}+\frac{n_{13}^{\prime}}{\gcd(N,r^{\prime})}w_{2}^{(N)}\cup
c_{1}+\frac{n_{23}^{\prime}}{\gcd(M,r)}w_{2}^{(M)}\cup c_{1}\right)~{},$ (113)

where the first line uses (100). Here, we have adopted a short hand notation
$\mathcal{P}$. For $\mathcal{P}(w^{(N)})$ with even $N$, $\mathcal{P}$ should
be understood as the Pontryagin square and $\mathcal{P}(w^{(N)})$ is a
$\mathbb{Z}_{2N}$-valued four-cocycle. For $\mathcal{P}(w^{(N)})$ with odd
$N$, it should be understood as $\mathcal{P}(w^{(N)})=w^{(N)}\cup w^{(N)}$,
which is a $\mathbb{Z}_{N}$-valued four-cocycle. $\mathcal{P}(w^{(M)})$ should
be interpreted similarly. Using the relation (101), we can express it as

$\displaystyle S_{\text{anom}}$
$\displaystyle=2\pi\int\left(\left(\frac{n_{1}^{\prime}}{2N}+\frac{n_{13}^{\prime}M/L}{\gcd(N,r^{\prime})}\right){\mathcal{P}}(w_{2}^{(N)})+\left(\frac{n_{2}^{\prime}}{2M}+\frac{n_{23}^{\prime}N/L}{\gcd(M,r^{\prime})}\right){\mathcal{P}}(w_{2}^{(M)})+\frac{n_{3}^{\prime}}{2r^{\prime}}c_{1}^{2}\right)$
(114)
$\displaystyle\quad+2\pi\int\left(\frac{n_{12}^{\prime}}{L}+\frac{n_{13}^{\prime}N/\gcd(N,r^{\prime})}{L}+\frac{n_{23}^{\prime}M/\gcd(M,r)}{L}\right)w_{2}^{(N)}\cup
w_{2}^{(M)}~{}.$ (115)

We can show that this anomaly can be expressed in terms of the statistics of
$v_{1}=u_{2}u_{3}^{N/L}$ and $v_{2}=u_{1}u_{3}^{M/L}$ which are given by
$\theta_{v_{1}}=e^{2\pi\mathrm{i}h_{v_{1}}}$,
$\theta_{v_{2}}=e^{2\pi\mathrm{i}h_{v_{2}}}$, and
$M_{v_{1},v_{2}}=e^{2\pi\mathrm{i}S_{12}}$ with

$\displaystyle
h_{v_{2}}=\frac{n_{1}^{\prime}}{2N}+\frac{n_{13}^{\prime}M/L}{\gcd(N,r^{\prime})}+\frac{n_{3}^{\prime}(M/L)^{2}}{2r^{\prime}},\quad
h_{v_{1}}=\frac{n_{2}^{\prime}}{2M}+\frac{n_{23}^{\prime}N/L}{\gcd(M,r^{\prime})}+\frac{n_{3}^{\prime}(N/L)^{2}}{2r^{\prime}},$
(116) $\displaystyle
S_{12}=\frac{n_{12}^{\prime}}{L}+\frac{n_{13}^{\prime}N/L}{\gcd(N,r^{\prime})}+\frac{n_{23}^{\prime}M/L}{\gcd(M,r^{\prime})}+\frac{n_{3}^{\prime}}{r^{\prime}}\frac{MN}{L^{2}}.$
(117)

We introduce $\mathbb{Z}$-valued cochains $\tilde{w}_{2}^{(Nr)}$ and
$\tilde{w}_{2}^{(Mr)}$

$\displaystyle\tilde{w}_{2}^{(Nr)}:=w_{2}^{(N)}+Nc_{1},\quad\tilde{w}_{2}^{(Mr)}:=w_{2}^{(M)}+Mc_{1},$
(118)

where we’ve taken lifts of $w_{2}^{(N)},w_{2}^{(M)}$ to integer cochains.
These integer lifts of $w_{2}^{(N)},w_{2}^{(M)}$ (still denoted using same
notations here) are chosen such that

$c_{1}=\frac{M}{L}w_{2}^{(N)}+\frac{N}{L}w_{2}^{(M)}~{}.$ (119)

We can then write the anomaly as

$\displaystyle S_{\text{anom}}$
$\displaystyle=2\pi\int\left(h_{v_{1}}{\mathcal{P}}\left(\tilde{w}_{2}^{(Nr)}\right)+h_{v_{2}}{\mathcal{P}}\left(\tilde{w}_{2}^{(Mr)}\right)+S_{12}\tilde{w}_{2}^{(Nr)}\cup\tilde{w}_{2}^{(Mr)}\right)+S_{\text{theta}}~{},$
(120) $\displaystyle S_{\text{theta}}$
$\displaystyle=-2\pi\int\left(\frac{n_{3}^{\prime}}{2r^{\prime}}{\mathcal{P}}(\frac{M}{L}w_{2}^{(N)}+\frac{N}{L}w_{2}^{(M)}+2\frac{MN}{L}c_{1})+\frac{n_{3}^{\prime}}{r^{\prime}}\left(\frac{MN}{L}\right)^{2}c_{1}^{2}+\frac{n_{1}^{\prime}N+n_{2}^{\prime}M}{2}c_{1}^{2}+\frac{n_{3}^{\prime}}{2r^{\prime}}c_{1}^{2}\right)$
(121)
$\displaystyle=-2\pi\int\left(\frac{n_{3}^{\prime}}{r^{\prime}}\left(1+\frac{2MN}{L}+\frac{3M^{2}N^{2}}{L^{2}}\right)+\frac{n_{1}^{\prime}N+n_{2}^{\prime}M}{2}\right)c_{1}^{2}~{}.$
(122)

Note that the last term $S_{\text{theta}}$ of $S_{\text{anom}}$ can be
rewritten as a term entirely proportional to $c_{1}^{2}$, which can be viewed
as a $\Theta$-term. Such a term which can be canceled by a local counterterm.
The anomaly does not depend on the choices of integer lifts of $w_{2}^{(N)}$
and $w_{2}^{(M)}$ as manifested in Eq. (114). After making such a choice, the
rewriting of $S_{\text{anom}}$ above shows that the anomaly only depends on
the statistics of $v_{1}$ and $v_{2}$. Thus. we can interpret the anomaly as
coming from condensing $v_{1},v_{2}$. Recall that $v_{1}$ and $v_{2}$ are
intrinsic to the symmetry fractionalization pattern $\mu$. Here, we’ve
introduced the integer lifts of $w_{2}^{(N)}$ and $w_{2}^{(M)}$ to argue that
the anomaly is intrinsic to the symmetry fractionalization pattern $\mu$. In
all of the following discussions, $w_{2}^{(N)}$ and $w_{2}^{(M)}$ should still
be viewed as $\mathbb{Z}_{N}$\- and $\mathbb{Z}_{M}$-valued two-cocycles.

To connect to the previous discussions in Sec. III.4, we need to consider a
value of $r$ that is a multiple of $r^{\prime}/\gcd(r^{\prime},MN/L)$ as
discussed above. $v_{1}$ and $v_{2}$ used in this subsection are identified
with $v_{1}$ and $v_{2}$ introduced in Sec. III.4.2.The parameters $n_{1}$,
$n_{2}$ and $n_{12}$ introduced in Eq. (58) corresponds to $h_{v_{1}}$,
$h_{v_{2}}$ and $S_{12}$ via $h_{v_{2}}=\frac{n_{2}}{2Nr}$,
$h_{v_{1}}=\frac{n_{1}}{2Mr}$, $S_{12}=\frac{n_{12}}{Lr}$:

$\displaystyle
n_{1}=2Mrh_{v_{1}}=rn_{2}^{\prime}+2rn_{23}^{\prime}\frac{M}{\gcd(M,r^{\prime})}\frac{N}{L}+n_{3}^{\prime}\frac{r}{r^{\prime}}\frac{MN^{2}}{L^{2}},$
(123) $\displaystyle
n_{2}=2Nrh_{v_{2}}=rn_{1}^{\prime}+2rn_{13}^{\prime}\frac{N}{\gcd(N,r^{\prime})}\frac{M}{L}+n_{3}^{\prime}\frac{r}{r^{\prime}}\frac{NM^{2}}{L^{2}},$
(124) $\displaystyle
n_{12}=LrS_{12}=rn_{12}^{\prime}+rn_{13}^{\prime}\frac{N}{\gcd(N,r^{\prime})}+rn_{23}^{\prime}\frac{M}{\gcd(M,r^{\prime})}+n_{3}^{\prime}\frac{r}{r^{\prime}}\frac{MN}{L}~{}.$
(125)

We want to exam if the bulk term $S_{\text{anom}}$ from the one-form anomaly
can be canceled by a “fractional-level” Chern-Simons counterterm $\left({\rm
SU}(N)_{k_{1}^{\prime}}\times{\rm
U}(1)_{k_{2}^{\prime}}\right)/(\mathbb{Z}_{N}\times\mathbb{Z}_{M})$ for the
$G={\rm U}(N)/\mathbb{Z}_{M}=\left({\rm SU}(N)\times{\rm
U}(1)\right)/(\mathbb{Z}_{N}\times\mathbb{Z}_{M})$ gauge field, which can be
written as correlated ${\rm SU}(N)/\mathbb{Z}_{N}$ gauge field and ${\rm
U}(1)/(\mathbb{Z}_{N}\times\mathbb{Z}_{M})={\rm
U}(1)/\mathbb{Z}_{MN/L}\cong{\rm U}(1)$ gauge field (the
$\mathbb{Z}_{N}\times\mathbb{Z}_{M}$ quotient generated by $g\sim
ge^{2\pi\mathrm{i}/N}g\sim e^{2\pi\mathrm{i}/M}g$ for an ${\rm U}(1)$ element
$g$ is equivalent to the $\mathbb{Z}_{MN/L}$ quotient $g\sim
e^{2\pi\mathrm{i}\left({t_{M}\over
N}+\frac{t_{N}}{M}\right)}g=e^{2\pi\mathrm{i}L/(MN)}g$ for integers
$t_{N},t_{M}$ that satisfy $t_{N}N+t_{M}M=L$). The action $S[A]$ of the
$G={\rm U}(N)/\mathbb{Z}_{M}$ gauge field is given by the two gauge fields as

$S[A]=k_{1}^{\prime}\text{CS}_{{\rm
PSU}(N)}+\frac{k_{2}^{\prime}L^{2}}{M^{2}N^{2}}\text{CS}_{{\rm U}(1)}~{},$
(126)

where $\text{CS}_{{\rm PSU}(N)}$ represents a Chern-Simons term for the ${\rm
PSU}(N)$ gauge field obtained from restricting the $G={\rm
U}(N)/\mathbb{Z}_{M}$ gauge field $A$ to the ${\rm PSU}(N)$ subgroup.
$\text{CS}_{{\rm U}(1)}$ represents the Chern-Simons term for the ${\rm U}(1)$
gauge field obtained from restricting the $G={\rm U}(N)/\mathbb{Z}_{M}$ gauge
field $A$ to the center ${\rm U}(1)$ subgroup. This action $S[A]$ physically
characterizes the $G={\rm U}(N)/\mathbb{Z}_{M}$ Hall response in the 2+1d
topological phase $\mathcal{C}$. We note that $k_{1}^{\prime},k_{2}^{\prime}$
in general can be any real numbers (they will be be chosen suitably in the
following anomaly considerations), which simply gives rise to the
$\Theta$-terms in the bulk. For our purpose of canceling the anomaly
contribution from $S_{\rm anom}$, we can take the local Chern-Simons
counterterms such that $k_{1}^{\prime}$ is an integer. Then while the ${\rm
SU}(N)_{k_{1}^{\prime}}$ Chern-Simons term does not depend on the bulk, the
${\rm PSU}(N)_{k_{1}^{\prime}}$ Chern-Simons term gives rise to the bulk term
$\int 2\pi k_{1}^{\prime}\frac{N-1}{2N}{\mathcal{P}}(w_{2}^{(N)})$ which can
be viewed as a $\Theta$-term with $\Theta$-angle $2\pi k_{1}^{\prime}$. Such a
term always evaluates to a multiple of $2\pi(N-1)/(2N)$. On top of the anomaly
Eq. (114) in the bulk, $S[A]$ contributes to the anomaly by

$S_{\rm
csc}=2\pi\int\left(k_{1}^{\prime}\frac{N-1}{2N}{\mathcal{P}}(w_{2}^{(N)})+{1\over
2}k_{2}^{\prime}{L^{2}\over M^{2}N^{2}}c_{1}^{2}\right)~{}.$ (127)

In the following we will examine whether the bulk topological term in Eq.
(114) can be canceled by a Chern-Simons counterterm $S[A]$ for the 0-form
symmetry $G={\rm U}(N)/\mathbb{Z}_{M}$. We will discuss the cases of even $MN$
(i.e. at least one of $M,N$ is even) and odd $MN$ (i.e. both $M,N$ are odd)
separately.

Case 1: even $MN$ \- Let us take the Chern-Simons counterterm $S[A]$ with
$k_{2}^{\prime}=-{n_{3}^{\prime}M^{2}N^{2}\over
L^{2}r^{\prime}}+\frac{MN}{L}k_{2}^{\prime\prime}$ for integer
$k_{2}^{\prime\prime}$. Using (101), we can write

$\displaystyle S_{\rm csc}=$ $\displaystyle
2\pi\int\left(k_{1}^{\prime}\frac{N-1}{2N}{\mathcal{P}}(w_{2}^{(N)})+\left({1\over
2}k_{2}^{\prime\prime}{L\over
MN}-\frac{n_{3}^{\prime}}{2r^{\prime}}\right)c_{1}^{2}\right)$ (128)
$\displaystyle=2\pi\int\left(k_{1}^{\prime}\frac{N-1}{2N}{\mathcal{P}}(w_{2}^{(N)})+k_{2}^{\prime\prime}\left(\frac{M}{2NL}{\mathcal{P}}(w_{2}^{(N)})+\frac{N}{2ML}{\mathcal{P}}(w_{2}^{(M)})+\frac{1}{L}w_{2}^{(M)}\cup
w_{2}^{(N)}\right)\right)-2\pi\frac{n_{3}^{\prime}}{2r^{\prime}}\int
c_{1}^{2}\;\text{ mod }2\pi~{}.$

Combining (114) and (128), we can first choose $k_{2}^{\prime\prime}$ so that
the $w_{2}^{(N)}\cup w_{2}^{(M)}$ terms are canceled. This cancellation
determines $k_{2}^{\prime\prime}$ up to $Lm$ for some $m\in\mathbb{Z}$. Then,
we choose $k_{1}^{\prime}$ to cancel the $\mathcal{P}(w_{2}^{(N)})$ term. The
total anomaly we are left with is given by

$S_{{\rm U}(N)/\mathbb{Z}_{M}\text{-SPT}}=S_{\rm csc}+S_{\rm
anom}=2\pi\left(\frac{n_{2}^{\prime}}{2M}-\frac{N/L}{2M}\cdot\left(mL+n_{12}^{\prime}+n_{13}^{\prime}\frac{N}{\gcd(N,r^{\prime})}-n_{23}^{\prime}\frac{M}{\gcd(M,r^{\prime})}\right)\right)\int{\mathcal{P}}(w_{2}^{(M)})~{}.$
(130)

which can be viewed as an effective action of a bulk 3+1d ${\rm
U}(N)/\mathbb{Z}_{M}$ SPT phase that describes the ’t Hooft anomaly of the
$G={\rm U}(N)/\mathbb{Z}_{M}$ 0-form symmetry on the boundary. By choosing a
suitable $m\in\mathbb{Z}$, and using the property that there exist integers
$t_{N}^{\prime},t_{M}^{\prime}$ such that
$Nt_{N}^{\prime}+2Mt_{M}^{\prime}=\gcd(2M,N)$, we can reduce the anomaly to

$S_{{\rm U}(N)/\mathbb{Z}_{M}\text{-SPT}}={2\pi\over
2M}\left(n_{2}^{\prime}-\frac{N}{L}\left(n_{12}^{\prime}+n_{13}^{\prime}\frac{N}{\gcd(N,r^{\prime})}-n_{23}^{\prime}\frac{M}{\gcd(M,r^{\prime})}\right)\text{
mod }\gcd(2M,N)\right)\int{\mathcal{P}}(w_{2}^{(M)})~{}.$ (131)

Note that making a choice of $m$ amounts to adding local counterterms to the
effective action and does not change the equivalence class of this 3+1d ${\rm
U}(N)/\mathbb{Z}_{M}$ symmetry SPT phase. In terms of the statistics of
$v_{1},v_{2}$ in (123), the 3+1d effective action that characterizes the
$G={\rm U}(N)/\mathbb{Z}_{M}$ ‘t Hooft anomaly in the 2+1d topological phase
$\mathcal{C}$ is given by

$S_{{\rm U}(N)/\mathbb{Z}_{M}\text{-SPT}}={2\pi\over
2M}\left(\frac{n_{1}}{r}-\frac{N}{L}\frac{n_{12}}{r}\text{ mod
}\gcd(2M,N)\right)\int{\mathcal{P}}(w_{2}^{(M)})~{},$ (132)

which agrees with the anomaly in (73).

We note that if $M$ is odd, $N$ needs to be even, and $N/L$ is even. In the
case, since the topological spin of $u_{2}$ is
$e^{2\pi\mathrm{i}n_{2}^{\prime}/(2M)}$, $n_{2}^{\prime}$ must be even for
$\mathcal{C}$ to be a bosonic system. Thus, the above action $S_{{\rm
U}(N)/\mathbb{Z}_{M}\text{-SPT}}$ in (131) is always a multiple of $2\pi/M$
when $M$ is odd, and thus the action $S_{{\rm
U}(N)/\mathbb{Z}_{M}\text{-SPT}}$ is well-defined.

When $M=1$, $N/L$ is even, $n_{2}^{\prime}=0$, and the anomaly is trivial,
which is in agreement with Sec. III.3.

For the case of $u_{1}=1,u_{3}=1,r=1$, Ref. [Benini  _et al._ , 2017]
discussed the anomaly on spin manifolds. Our discussion above agrees with the
result of Ref. [Benini  _et al._ , 2017] and is not limited to spin manifolds.

The total bulk topological term $S_{{\rm U}(N)/\mathbb{Z}_{M}\text{-SPT}}$
represents a trivial ’t Hooft anomaly if and only if

$\ell:=n_{2}^{\prime}-\frac{N}{L}\left(n_{12}^{\prime}+n_{13}^{\prime}\frac{N}{\gcd(N,r^{\prime})}-n_{23}^{\prime}\frac{M}{\gcd(M,r^{\prime})}\right)=0\quad\text{mod}\gcd(2M,N)~{}.$
(133)

When the anomaly is trivial, the fractional part of the quantum Hall response
is

$\displaystyle\sigma_{H}^{(2)}=-\frac{L^{2}}{M^{2}N^{2}}k_{2}^{\prime}=+\frac{n_{3}^{\prime}}{r^{\prime}}+\frac{L}{MN}\left({\ell\over\gcd(2M,N)}t^{\prime}_{N}+n_{12}^{\prime}+n_{13}^{\prime}\frac{N}{\gcd(N,r^{\prime})}+n_{23}^{\prime}\frac{M}{\gcd(M,r^{\prime})}\right)$
(134)
$\displaystyle\sigma_{H}^{(1)}=-k_{1}^{\prime}=\left\\{\begin{array}[]{cc}-(N+1)\left(n_{1}+\frac{2n_{13}^{\prime}MN/L}{\gcd(N,r^{\prime})}-\frac{M}{L}\left({\ell\over\gcd(2M,N)}t^{\prime}_{N}+n_{12}^{\prime}+n_{13}^{\prime}\frac{N}{\gcd(N,r^{\prime})}+n_{23}^{\prime}\frac{M}{\gcd(M,r^{\prime})}\right)\right)&N\text{
even}\\\
-\left(n_{1}+\frac{2n_{13}^{\prime}MN/L}{\gcd(N,r^{\prime})}-\frac{M}{L}\left({\ell\over\gcd(2M,N)}t^{\prime}_{N}+n_{12}^{\prime}+n_{13}^{\prime}\frac{N}{\gcd(N,r^{\prime})}+n_{23}^{\prime}\frac{M}{\gcd(M,r^{\prime})}\right)\right)&N\text{
odd}\end{array}\right.\,,$ (137)

where we note that when $N$ is odd, both $M$ and $n_{1}^{\prime}$ are even. We
can stack an additional layer of integer quantum Hall response labeled by
integers $p,p^{\prime}$. This stacking corresponds to shifting
$\ell\rightarrow\ell+p^{\prime}\gcd(2M,N)$, and then
$k_{1}^{\prime}\rightarrow k_{1}^{\prime}+2Np$ (for even $N$) or
$k_{1}^{\prime}\rightarrow k_{1}^{\prime}+Np$ (for odd $N$) Here, by integer
quantum Hall response, we refer to a ${\rm U}(N)/\mathbb{Z}_{M}$ Chern-Simons
effective response action with properly quantized levels.

Case 2: odd $MN$ \- Let us take the Chern-Simons counterterm $S[A]$ with
$k_{2}^{\prime}=-{n_{3}^{\prime}M^{2}N^{2}\over
L^{2}r^{\prime}}+\frac{MN}{L}\left(1+\frac{MN}{L}\right)k_{2}^{\prime\prime}$
for an integer $k_{2}^{\prime}$. Using (101), we can write

$\displaystyle S_{\text{csc}}=$ $\displaystyle
2\pi\int\left(k_{1}^{\prime}\frac{N-1}{2N}{\mathcal{P}}(w_{2}^{(N)})+\left({1\over
2}\left(1+\frac{MN}{L}\right)k_{2}^{\prime\prime}{L\over
MN}-\frac{n_{3}^{\prime}}{2r^{\prime}}\right)c_{1}^{2}\right)$ (138)
$\displaystyle=2\pi\int\left(k_{1}^{\prime}\frac{N-1}{2N}{\mathcal{P}}(w_{2}^{(N)})+k_{2}^{\prime\prime}\left(\left(1+\frac{MN}{L}\right)\frac{M}{2NL}{\mathcal{P}}(w_{2}^{(N)})+\left(1+\frac{MN}{L}\right)\frac{N}{2ML}{\mathcal{P}}(w_{2}^{(M)})\right)\right)$
(139) $\displaystyle+2\pi\int k_{2}^{\prime\prime}\frac{1}{L}w_{2}^{(M)}\cup
w_{2}^{(N)}-2\pi\frac{n_{3}^{\prime}}{2r^{\prime}}\int c_{1}^{2}\quad\text{
mod }2\pi~{},$ (140)

where we note that the coefficient in front of ${\mathcal{P}}(w_{2}^{(N)})$ is
always an integer multiple of $1/N$, and similarly the coefficient in front of
${\mathcal{P}}(w_{2}^{(M)})$ is always an integer multiple of $1/M$.

Combining (114) and (138), we can first choose $k_{2}^{\prime\prime}$ to
cancel the term $w_{2}^{(N)}\cup w_{2}^{(M)}$ (with freedom $Lm$ for
$m\in\mathbb{Z}$), and then choose $k_{1}^{\prime}$ to cancel the term
$\mathcal{P}(w_{2}^{(N)})$. Then, we are left with the 3+1d effective action

$\displaystyle S_{{\rm U}(N)/\mathbb{Z}_{M}\text{-SPT}}=S_{\rm csc}+S_{\rm
anom}$
$\displaystyle~{}=2\pi\left(\frac{n_{2}^{\prime}}{2M}+\frac{n_{23}^{\prime}N/L}{\gcd(M,r^{\prime})}-\frac{N/L}{2M}\left(1+\frac{MN}{L}\right)\cdot\left(mL+n_{12}^{\prime}+n_{13}^{\prime}\frac{N}{\gcd(N,r^{\prime})}+n_{23}^{\prime}\frac{M}{\gcd(M,r^{\prime})}\right)\right)\int{\mathcal{P}}(w_{2}^{(M)})~{}.$
(141)

By choosing a suitable $m\in\mathbb{Z}$, we can reduce $S_{{\rm
U}(N)/\mathbb{Z}_{M}\text{-SPT}}$ to

$\displaystyle S_{{\rm U}(N)/\mathbb{Z}_{M}\text{-SPT}}$
$\displaystyle={2\pi\over
2M}\left(n_{2}^{\prime}+2n_{23}^{\prime}\frac{MN}{L\gcd(M,r^{\prime})}-\frac{N}{L}\left(1+\frac{MN}{L}\right)\left(n_{12}^{\prime}+\frac{n_{13}^{\prime}N}{\gcd(N,r^{\prime})}+\frac{n_{23}^{\prime}M}{\gcd(M,r^{\prime})}\right)\text{
mod }(2L)\right)\int{\mathcal{P}}(w_{2}^{(M)})~{}.$ (142)

Note that since $n_{2}^{\prime}$ is even (due to the consistency of the
topological spin of $u_{2}$ in a bosonic system), we have

$\frac{MN}{L}\left(n_{2}^{\prime}+2n_{23}^{\prime}\frac{MN/L}{\gcd(M,r^{\prime})}\right)=0\text{
mod }2L~{},$ (143)

Thus we can rewrite the 3+1d effective action $S_{{\rm
U}(N)/\mathbb{Z}_{M}\text{-SPT}}$, which characterizes the ‘t Hooft anomaly in
the 2+1d topological phase $\mathcal{C}$, as

$\displaystyle S_{{\rm U}(N)/\mathbb{Z}_{M}\text{-SPT}}$
$\displaystyle={2\pi\over
2M}\left(1+\frac{MN}{L}\right)\left(n_{2}^{\prime}+2n_{23}^{\prime}\frac{MN}{L\gcd(M,r^{\prime})}-\frac{N}{L}\left(n_{12}^{\prime}+\frac{n_{13}^{\prime}N}{\gcd(N,r^{\prime})}+\frac{n_{23}^{\prime}M}{\gcd(M,r^{\prime})}\right)\text{
mod }(2L)\right)\int{\mathcal{P}}(w_{2}^{(M)})~{}.$ (144)

Note that the $\left(1+MN/L\right)$ prefactor can be replaced by $(1+M)$.
Since $N/L$ is odd, the difference between the prefactors
$\left(1+MN/L\right)$ and $\left(1+M\right)$ is a multiple of $2M$. This
replacement keeps the effective action $S_{{\rm
U}(N)/\mathbb{Z}_{M}\text{-SPT}}$ invariant modulo $2\pi$. Written in terms of
the statistics of $v_{1},v_{2}$ in (123), the 3+1d effective action that
captures the ‘t Hooft anomaly is given by

$S_{{\rm U}(N)/\mathbb{Z}_{M}\text{-SPT}}={2\pi\over
M}\frac{1+M}{2}\left(\frac{n_{1}}{r}-\frac{N}{L}\frac{n_{12}}{r}\text{ mod
}(2L)\right)\int{\mathcal{P}}(w_{2}^{(M)})~{},$ (145)

which agrees with the result in (77).

For the case of $u_{1}=1,u_{3}=1,r=1$, Ref. [Benini  _et al._ , 2017]
discussed the anomaly on spin manifolds. Our discussion above agrees with the
result of Ref. [Benini  _et al._ , 2017] and is not limited to spin manifolds.

In general, the ${\rm U}(N)/\mathbb{Z}_{M}$ ’t Hooft anomaly vanishes if and
only if

$\ell:=n_{2}^{\prime}+2n_{23}^{\prime}\frac{MN}{L\gcd(M,r^{\prime})}-\frac{N}{L}\left(1+\frac{MN}{L}\right)\left(n_{12}^{\prime}+\frac{n_{13}^{\prime}N}{\gcd(N,r^{\prime})}+\frac{n_{23}^{\prime}M}{\gcd(M,r^{\prime})}\right)=0\quad\text{mod
}2L~{}.$ (146)

When the term can be canceled, the fractional part of the quantum Hall
response is

$\displaystyle\sigma_{H}^{(2)}=-\frac{L^{2}}{M^{2}N^{2}}k_{2}^{\prime}=\frac{n_{3}^{\prime}}{r^{\prime}}+\left(1+\frac{L}{MN}\right)\left({\ell\over
L}t_{N}+n_{12}^{\prime}+n_{13}^{\prime}\frac{N}{\gcd(N,r^{\prime})}+n_{23}^{\prime}\frac{M}{\gcd(M,r^{\prime})}\right)$
(147)
$\displaystyle\sigma_{H}^{(1)}=-k_{1}^{\prime}=-\left(n_{1}+\frac{2n_{13}^{\prime}MN/L}{\gcd(N,r^{\prime})}-\frac{M}{L}\left(1+\frac{MN}{L}\right)\left({\ell\over
L}t_{N}+n_{12}^{\prime}+n_{13}^{\prime}\frac{N}{\gcd(N,r^{\prime})}+n_{23}^{\prime}\frac{M}{\gcd(M,r)}\right)\right)~{},$
(148)

Note that when $N$ is odd, $n_{1}^{\prime}$ must be even. We can stack
additional integer quantum Hall response labeled by integers $p,p^{\prime}$
that corresponds to shifting $\ell\rightarrow\ell+2p^{\prime}L$, and then
$k_{1}^{\prime}\rightarrow k_{1}^{\prime}+Np$.

###  IV.5 A gapless example

Consider ${\rm SU}(N)_{k}$ with $N_{f}$ massless fermions in the adjoint
representation of ${\rm SU}(N)$. For $N_{f}=1$, the phase diagram of the
theory is proposed in Ref. [Gomis  _et al._ , 2018] and it is believed to have
a critical point. For $k<N/2$, this critical point is described by the
Goldstino that results from the spontaneously broken ${\mathcal{N}}=1$
supersymmetryWitten (2000). The discussion below applies to general $N_{f}$.

The theory has $\mathbb{Z}_{N}$ center one-form symmetry that acts on the
${\rm SU}(N)$ fundamental Wilson line. Let us enrich the theory with a $G={\rm
U}(1)$ 0-form global symmetry, by coupling the theory to the following
$\mathbb{Z}_{N}$ two-form gauge field $B$ for the one-form $\mathbb{Z}_{N}$
symmetry

$B=qc_{1}\text{ mod }N~{},$ (149)

where $q$ is an integer, and $c_{1}$ is the first Chern class of the
background ${\rm U}(1)$ gauge field. Without loss of generality, we can take
$q$ to be a divisor of $N$. This means that the fundamental Wilson line
carries fractional ${\rm U}(1)$ charge $q/N$. This 0-form ${\rm U}(1)$
symmetry does not act on the local operators. One can extend the symmetry
group $G$ to $\tilde{G}=\tilde{\rm U}(1)$, which is the $r$-fold covering of
${\rm U}(1)$, with $r=N/q$ (Here, we take $q$ to be a divisor of $N$ for
simplicity).

The one-form $\mathbb{Z}_{N}$ symmetry is anomalous. The anomaly can be
calculated from the bare Chern-Simons term ${\rm SU}(N)_{k+NN_{f}/2}$ obtained
by giving the fermions a mass. In the presence of background two-form gauge
field $B$, the ${\rm SU}(N)$ gauge field becomes a ${\rm PSU}(N)$ gauge field,
and the ${\rm SU}(N)$ Chern-Simons term becomes a ${\rm PSU}(N)$ Chern-Simons
term. This ${\rm PSU}(N)$ Chern-Simons term is not well-defined by itself in
2+1d. It can be viewed as the boundary of a 3+1d bulk with an effective action

$2\pi\frac{(k+NN_{f}/2)(N-1)}{2N}\int{\mathcal{P}}(B)~{}.$ (150)

For the configuration $B=qc_{1}$, this bulk term can be canceled, via Eq.
(80), by the Chern-Simons term

$S[A]=\frac{-q^{2}(k+NN_{f}/2)(N-1)/N+k^{\prime}}{4\pi}AdA~{},$ (151)

where $A$ is the ${\rm U}(1)$ gauge field, and $k^{\prime}$ is an integer that
represents a properly quantized Chern-Simons term
$\frac{k^{\prime}}{4\pi}AdA$. This cancellation implies that anomaly for the
one-form symmetry does not induce any non-trivial ’t Hooft anomaly for the
0-form global symmetry $G={\rm U}(1)$.

In terms of the $\tilde{G}=\tilde{{\rm U}}(1)$ gauge field $\tilde{A}$ related
to $A$ by $A=r\tilde{A}=\frac{N}{q}\tilde{A}$, $S[A]$ can be written as a
$\tilde{\rm U}(1)_{\kappa}$ Chern-Simons term with

$\kappa=-N(N-1)(k+NN_{f}/2)+k^{\prime}\frac{N^{2}}{q^{2}}~{}.$ (152)

Gauging the 0-form global symmetry $G={\rm U}(1)$ produces the theory

${{\rm SU}(N)_{k}\times\tilde{\rm
U}(1)_{\kappa}\over\mathbb{Z}_{r}}+N_{f}\text{ adjoint fermions}~{},$ (153)

where the adjoint fermions do not carry any gauge charge under $\tilde{\rm
U}(1)$.

##  V Conclusion and Discussion

In this work we develop a general procedure for gauging a compact, connected
Lie group symmetry in a 2+1d topological phase. The procedure can be
formulated algebraically using the data of the underlying MTC and the symmetry
fractionalization class. We also provide a field theory derivation for the
result, which can be applied to certain gapless theories as well.

While we focus on gauging of the 2+1d topological phase, there is a closely
related construction for chiral 1+1d conformal field theories (CFT) on the
boundary. Gauging a finite group symmetry is equivalent to orbifolding the
boundary CFT. When the CFT has Lie group symmetry, e.g. Wess-Zumino-Witten
theories, gauging a Lie subgroup is essentially equivalent to the coset
construction Nahm (1987); Gawedzki and Kupiainen (1989); Bardakci  _et al._
(1988). Thus our results can be understood as a general categorical
description of cosets. For instance, take $G_{k}$ Wess-Zumino-Witten model for
simply connected $G$ and gauge a Lie group symmetry $H$ gives the coset
$(G/H)_{k}$. If $G,H$ have common center $C$, the representations of the
current algebra are that of $G,H$ subject to selection rule, and this
corresponds to a bulk $(G\times H)/C$ Chern-Simons theory Moore and Seiberg
(1989), which is related to the $G$ Chern-Simons theory corresponding to the
$G_{k}$ Wess-Zumino-Witten model by gauging an $H/C$ symmetry.

The gauged theory can in principle have an emergent 0-form
$\widehat{\pi_{1}(G)}=\text{Hom}(\pi_{1}(G),{\rm U}(1))$ symmetry for the
monopoles of the dynamical $G$ gauge field. The monopoles are classified by
$\pi_{1}(G)$, Preskill (1984) and the associated symmetry is its Pontryagin
dual $\widehat{\pi_{1}(G)}$. In the general gauging procedure discussed in the
preceding sections, we have focused on the topological order of the resulting
theory, and have not considered how the dual symmetry acts. In the case of
$G={\rm U}(1)$, aspects of the dual symmetry, which is also ${\rm U}(1)$ in
this case, on the gauged theory are discussed in App. C and in Ref. [Cheng and
Jian, 2022]. It is particularly relevant when before gauging the system has
$\sigma_{H}=0$. Then after gauging, if the dual U(1) symmetry is present, it
can become spontaneously broken leading to Goldstone modes.

Let us now briefly discuss what happens when $G$ is a simple Lie group. In
this case, the dual symmetry group $\widehat{\pi_{1}(G)}$ is a finite Abelian
group. We will assume that it is present, namely not explicitly broken, after
gauging, i.e. we do not include deformation by the topologically non-trivial
monopole operators of the $G$ gauge field to break the dual symmetry
explicitly. In the simplest case, suppose that we gauge the $G$ symmetry in a
completely trivial gapped state (in the trivial invertible phase with $G$
symmetry). The gauging procedure gives the theory $\tilde{G}_{0}/\pi_{1}(G)$.
Here we take $\tilde{G}$ to be the universal cover of $G$, and $K=\pi_{1}(G)$.
$\tilde{G}_{0}$ stands for the standard Yang-Mills theory of gauge group
$\tilde{G}$ (where the subscript “0” simply means that the Chern-Simons level
is $0$). Also, we assume here that the matter fields in the original trivial
gapped states remains gapped in the gauging process. Since $\tilde{G}$ is
simply-connected, the $\tilde{G}$ Yang-Mills theory is in a completely
featureless confined phase. Now the second step of gauging the center 1-form
$\pi_{1}(G)$ symmetry leads to the spontaneous breaking of the magnetic 0-form
symmetry $\widehat{\pi_{1}(G)}$, giving rise to $|\pi_{1}(G)|$ vacua on
$S^{2}$. 999 This follows from the fact that gauging the one-form symmetry
$\pi_{1}(G)$ in the trivial theory produces free two-form gauge theory with
Abelian gauge group $\pi_{1}(G)$, which can be dualized to topological point
operators valued in $\pi_{1}(G)$, whose value labels the vacua.  When
$\sigma_{H}\neq 0$, the gauged theory $\mathcal{D}$ is obtained from gauging
the 1-form $K$ symmetry in $\mathcal{C}\boxtimes\tilde{G}_{-\sigma_{H}}$.
Therefore, the resulting topological phase is enriched by the magnetic 0-form
$\hat{K}=\widehat{\pi_{1}(G)}$ symmetry. The action of $\hat{K}$ in
$\mathcal{D}$ is generally complicated, but it can be described in the
following way: gauging the $\hat{K}$ symmetry in $\mathcal{D}$ should give
back the $\mathcal{C}\boxtimes\tilde{G}_{-\sigma_{H}}$ theory. This fact
completely determines how $\hat{K}$ acts in the $\mathcal{D}$ theory. To give
a simple example, let us consider $\mathcal{C}=\mathrm{SU}(2)_{2}$ with
$\sigma_{H}=2$. Then gauging produces
$[\mathrm{SU}(2)_{2}\boxtimes\mathrm{SU}(2)_{-2}]/\mathbb{Z}_{2}$, which has
the topological order of the $\mathbb{Z}_{2}$ toric code. The generator of the
magnetic symmetry $\pi_{1}(\mathrm{SO}(3))=\mathbb{Z}_{2}$ acts as the
electro-magnetic duality symmetry in the toric code (with stacking of a non-
trivial $\mathbb{Z}_{2}$ SPT phase, but the symmetry is not fractionalized on
the particles).

We discuss some directions for future work.

In the finite group case, gauging the symmetry is done first by introducing
symmetry defects and constructing a $G$-crossed braided tensor category, and
then taking the equivariantization (i.e. projecting to $G$-invariant space) to
obtain the gauged theory. Generalization of $G$-crossed braided tensor
category for Lie group symmetry is challenging, but for our purpose it is not
necessary. In our approach we sidestep the intermediate defect theory by using
the group extension $\tilde{G}$. By construction $\tilde{G}$ has no
fractionalization and thus gauging $\tilde{G}$ simply gives a decoupled Chern-
Simons theory, whose topological order is well-understood. It is an
interesting question to see how the two approaches are related. From our two-
step gauging procedure, the issue essentially reduces to understanding
$\tilde{G}$ defects in a $\tilde{G}$ SPT phase and how its gauging leads to
$\tilde{G}$ Chern-Simons theory, from a purely algebraic viewpoint. This is an
interesting question for future work.

In this work, we have considered general compact connected Lie groups. An
important consequence of the connected-ness is that the symmetry can not
permute anyons, which greatly simplifies gauging. When the Lie group has
multiple connected components, e.g. the O$(N)$ group, the 0-form symmetry,
such as the reflection in O$(N)$, can also permute the anyons. We now briefly
outline how our gauging procedure can be generalized to a Lie group $G$ with
multiple components. For such a Lie group, the identity component $G_{0}$
(i.e. the connected component that contains the identity) is a normal
subgroup, and we have the following group extension:

$1\rightarrow G_{0}\rightarrow G\rightarrow G/G_{0}=H\rightarrow 1.$ (154)

Here the quotient $H$ is a finite group. For an example, if $G={\rm O}(N)$,
then $G_{0}={\rm SO}(N),H=\mathbb{Z}_{2}$, and indeed we have

${\rm O}(N)=\begin{cases}{\rm SO}(N)\times\mathbb{Z}_{2}&N\text{ odd}\\\ {\rm
SO}(N)\rtimes\mathbb{Z}_{2}&N\text{ even}\end{cases}.$ (155)

Gauging $G$ can be carried out in two steps. First, we can gauge the identity
component $G_{0}$. This can be done using the two-step gauging procedure
developed in this work. Then once $G_{0}$ is gauged, $H$ remains a global
symmetry in the resulting topological phase and can be further gauged
following the established procedure Barkeshli  _et al._ (2019).

Now there is a new ingredient in the ’t Hooft anomaly. The anomaly of $G_{0}$
can be derived as before from the obstruction in gauging $G_{0}$. If $G_{0}$
is non-anomalous and can be consistently gauged, we need to check whether $H$
remains a good symmetry in the gauged theory. If not, then there must be a
mixed anomaly between $G_{0}$ and $H$. In fact, a more careful analysis shows
that when there is a mixed anomaly, gauging $G_{0}$ results in the extension
of $H$. Depending on the nature of the anomaly, $H$ may be extended to a
larger 0-form symmetry group, or a two-group Cheng  _et al._ .

We note that the symmetry fractionalization pattern
$\mathcal{H}^{2}(G,\mathcal{A})=H^{2}(\text{B}G,\mathcal{A})$ can be formally
viewed as classifying families of gapped states in the same topological phase
over the parameter space B$G$. The classifying space B$G$ here can be replaced
by any smooth closed manifold $X$, where $H^{2}(X,\mathcal{A})$ partially
classifies family of topological phases over the parameter space $X$ Kitaev
(2006); Aasen  _et al._ (2022). This information also provides a way to couple
the topological quantum field theory to a sigma model with $X$ as the target
space.101010 We thank Anton Kapustin for discussing on this point. . This is
relevant when a global continuous symmetry is spontaneously broken in a
topological phase (e.g. quantum Hall ferromagnet), and the topological field
theory is coupled to the Goldstone modes Else (2021).

In our construction, we couple the theory to $G=\tilde{G}/K$ gauge field,
where the quotient can be interpreted as coupling to $\tilde{G}$ gauge field
and then gauging a $K$ one-form symmetry. Instead of gauging a relativistic
one-form symmetry, we can gauge a subsystem one-form symmetry that acts only
along a direction, by introducing foliated gauge field $B^{k}$ that obeys the
relation $B^{k}e^{k}=0$ with respect to foliation one-form $e^{k}$ (for
instance, see Ref. [Hsin and Slagle, 2021]). We can use such procedure to
construct fracton phases in general higher dimensions.

## Acknowledgment

We thank Anton Kapustin and Zohar Komargodski for discussions. We thank
Maissam Barkeshli, Shu-Heng Shao and Zhenghan Wang for comments on this
manuscript. P.-S. H. is supported by the Simons Collaboration on Global
Categorical Symmetries. M.C. acknowledges support from NSF under award number
DMR-1846109. C.-M. J. is supported by a faculty startup grant at Cornell
University.

##  Appendix A One-form symmetry anomaly in 2+1d

Let us review the anomaly for one-form symmetry in 2+1d. The symmetry is
generated by topological line operators that obey an Abelian fusion rule. Let
us denote the set of such topological line operators by ${\mathcal{A}}$, they
generate the intrinsic one-form symmetry of the theory. Such line operators
can have non-trivial braiding statistics, and open lines create Abelian anyons
in the theory. When the statistics is non-trivial, the symmetry cannot be
gauged, and there is an ’t Hooft anomaly for the one-form symmetry Hsin  _et
al._ (2019). We note that if the Abelian anyons are bosons, the corresponding
one-form symmetry can be gauged, which is in effect the same as condensing the
Abelian anyons. The statistics of the Abelian anyons can be encoded in a
quadratic function $h:{\mathcal{A}}\rightarrow{\rm
U}(1)\cong\mathbb{R}/\mathbb{Z}$ that gives the spin of the Abelian anyons.
For $a\in{\mathcal{A}}$, the topological spin is $\theta_{a}=e^{2\pi ih[a]}$.
Denote the background two-form gauge field for the one-form symmetry by
$B^{\text{int}}$. The anomaly can be described by the 3+1d SPT phase with one-
form symmetry given by the effective action Hsin  _et al._ (2019)

$S_{\text{anom}}=2\pi\int h[B^{\text{int}}]~{}.$ (156)

Explicitly, if ${\mathcal{A}}=\prod\mathbb{Z}_{N_{i}}$, denote the component
of an Abelian anyon $a$ by $a_{i}\in\mathbb{Z}_{N_{i}}$, the quadratic
function can be parameterized by integers $p_{ij}$ as

$h[a]=\sum_{i}\frac{p_{ii}}{2N_{i}}a_{i}^{2}+\sum_{i<j}\frac{p_{ij}}{\gcd(N_{i},N_{j})}a_{i}a_{j}\quad\text{mod
}1~{}.$ (157)

The 3+1d effective action is

$S_{\text{anom}}=2\pi\int\left(\sum_{i}\frac{p_{ii}}{2N_{i}}{\mathcal{P}}\left((B^{\text{int}})_{i}\right)+\sum_{i<j}\frac{p_{ij}}{\gcd(N_{i},N_{j})}(B^{\text{int}})_{i}\cup(B^{\text{int}})_{j}\right)~{},$
(158)

where $(B^{\text{int}})_{i}$ is the component of $B^{\text{int}}$ for the
$\mathbb{Z}_{N_{i}}$ factor of ${\mathcal{A}}$, and ${\mathcal{P}}$ is the
Pontryagin square operation:

${\mathcal{P}}(x)=\begin{cases}\tilde{x}\cup\tilde{x}-\tilde{x}\cup_{1}\delta\tilde{x}\text{
mod }2N_{i}&N_{i}\text{ even}\\\ x\cup x\text{ mod }N_{i}&N_{i}\text{
odd}\end{cases}$ (159)

Here $\tilde{x}$ is an integral lift of $x$. Note that when $N_{i}$ is odd,
$p_{i}$ must be even.

More generally, we can couple the theory to $K$-valued two-form gauge field
$B$ by the homomorphism $v:K\rightarrow{\mathcal{A}}$, by setting Benini  _et
al._ (2019)

$B^{\text{int}}=v(B)~{}.$ (160)

Then the anomaly for the $K$ one-form symmetry, which is an extrinsic symmetry
compared to the intrinsic one-form symmetry ${\mathcal{A}}$, is given by the
bulk effective action

$S_{\text{anom}}=2\pi\int h[v(B)]~{}.$ (161)

##  Appendix B Projective representation and universal cover

The elements of $\tilde{G}$, which is obtained from the central extension Eq.
(5), can be described by $(k,g)\in K\times G$ with the multiplication law

$(k,g)\cdot(k^{\prime},g^{\prime})=(kk^{\prime}\mu(g,g^{\prime}),gg^{\prime})~{},$
(162)

where $\mu\in\mathcal{H}^{2}(G,K)$ is a $K$-valued two-cocycle of group $G$.
For linear representation $\tilde{R}_{a}$ of $\tilde{G}$,

$\tilde{R}_{a}(1,g)\cdot\tilde{R}_{a}(1,g^{\prime})=\tilde{R}_{a}((\mu(g,g^{\prime}),1)(1,gg^{\prime}))=\tilde{R}_{a}(\mu(g,g^{\prime}))\tilde{R}_{a}(1,gg^{\prime})~{},$
(163)

where $\tilde{R}_{a}(\mu(g,g^{\prime}))$ is a ${\rm U}(1)$ phase since the
extension is central and $(\mu(g,g^{\prime}),1)$ is a central element of
$\tilde{G}$. Thus, $\tilde{R}_{a}\circ\mu$ can be viewed as the projective
2-cocycle of the projective representation of $G$, i.e.
$\tilde{R}_{a}\circ\mu\in\mathcal{H}^{2}(G,{\rm U}(1))$. The homomorphism
$q_{a}:K\rightarrow{\rm U}(1)$ essentially induces the map from the $K$-valued
cocycle $\mu$ to the ${\rm U}(1)$-valued cocycle $q_{a}(\mu)\equiv
R_{a}\circ\mu$.

Consider the fusion of two linear representations
$\tilde{R}_{a},\tilde{R}_{b}$ of $\tilde{G}$. They satisfy

$\displaystyle\tilde{R}_{a}\otimes\tilde{R}_{b}(1,g)\cdot\tilde{R}_{a}\otimes\tilde{R}_{b}(1,g^{\prime})=$
$\displaystyle\tilde{R}_{a}((\mu(g,g^{\prime})(1,gg^{\prime}))\otimes
R_{b}((\mu(g,g^{\prime})(1,gg^{\prime}))$ (164) $\displaystyle=$
$\displaystyle\tilde{R}_{a}(\mu(g,g^{\prime}))\tilde{R}_{b}(\mu(g,g^{\prime}))\left(\tilde{R}_{a}\otimes\tilde{R}_{b}(1,gg^{\prime})\right)~{},$
(165)

In the last line, we’ve used the fact that the extension is central and,
consequently, $\tilde{R}_{a}(\mu(g,g^{\prime}))$ and
$\tilde{R}_{b}(\mu(g,g^{\prime}))$ are ${\rm U}(1)$ phases proportional to the
identity matrix. Thus, for central extensions,

$\tilde{R}_{a}(\mu(g,g^{\prime}))\cdot\tilde{R}_{b}(\mu(g,g^{\prime}))=\left(\tilde{R}_{a}\otimes\tilde{R}_{b}\right)(\mu(g,g^{\prime}))~{}.$
(166)

This reproduces the relation (2). The discussion can be generalized to general
group extension. Benini  _et al._ (2018)

##  Appendix C SL$(2,\mathbb{Z})$ action for ${\rm U}(1)$ 0-form symmetry

For 2+1d topological phases with ${\rm U}(1)$ 0-form symmetry, we can perform
SL$(2,\mathbb{Z})$ transformation to map one topological phase to
another.Witten (2003). ${\mathcal{S}},{\mathcal{T}}$ are the generators of the
general SL$(2,\mathbb{Z})$ transformation. Denote the partition function of a
2+1d topological phase coupled to a ${\rm U}(1)$ background gauge field $A$ as
$Z[A]$. Each of the $\mathcal{S}$ and $\mathcal{T}$ operations transforms the
topological phase to a new one whose partition is given by

${\mathcal{T}}:\;Z[A]\rightarrow Z^{\prime}[A]=Z[A]e^{\frac{i}{4\pi}\int
AdA},\quad{\mathcal{S}}:\;Z[A]\rightarrow
Z^{\prime}[A]=\sum_{a}Z[a]e^{\frac{i}{2\pi}\int adA}~{},$ (167)

where $\sum_{a}$ represents the path integral over the dynamical gauge field
$a$. The $\mathcal{S}$ operation represents gauging the original ${\rm U}(1)$
symmetry without adding additional local counterterm. The gauged theory has a
dual ${\rm U}(1)^{\prime}$ symmetry.

We note that for bosonic theories, the operation
${\mathcal{S}},{\mathcal{T}}^{2}$ are well-defined, while ${\mathcal{T}}$ is
not. Here, we will use the method of Sec. IV to consider general bosonic or
fermionic theories, where ${\mathcal{S}},{\mathcal{T}}$ are both well-defined
operations.

Let us start with a theory ${\mathcal{C}}$ with $\mathbb{Z}_{r}$ subgroup one-
form symmetry, generated by a line operator of topological spin
$\theta_{v}=e^{2\pi i\frac{n}{2r}}$ for some integer $n$. Then we can gauge
the ${\rm U}(1)$ symmetry to obtain

$\left({\mathcal{S}}{\mathcal{T}}^{k}\right){\mathcal{C}}=\frac{{\mathcal{C}}\times{\rm
U}(1)_{-nr+kr^{2}}}{\mathbb{Z}_{r}}~{},$ (168)

where we take the Hall response in the theory ${\mathcal{C}}$ to be $-n/(2\pi
r)$ by choosing suitable $n$ (which is now not just defined mod $2r$, but an
integer). The ${\mathcal{T}}^{k}$ operation changes the Chern-Simons
counterterm to $\frac{k-n/r}{4\pi}AdA$. We note that the theory
$\left({\mathcal{S}}{\mathcal{T}}^{k}\right){\mathcal{C}}$ has
$\mathbb{Z}_{|-n+kr|}$ one-form symmetry generated by the Wilson line
$v^{\prime}$ of ${\rm U}(1)$ charge $r$, which has topological spin
$\theta_{v^{\prime}}=e^{2\pi\mathrm{i}\frac{r}{2(-n+kr)}}$.

For our purpose, the transformation is
PSL$(2,\mathbb{Z})=$SL$(2,\mathbb{Z})/\mathbb{Z}_{2}$, where the quotient is
generated by ${\mathcal{S}}^{2}$. There are two relations to check using our
construction for gauging the U(1) symmetry: (1) ${\mathcal{S}}^{2}=1$ and (2)
$({\mathcal{S}\mathcal{T}})^{3}=1$. That means both of ${\mathcal{S}}^{2}=1$
and $({\mathcal{S}\mathcal{T}})^{3}=1$ map a 2+1d topological phase with a
0-form U(1) symmetry back to itself.

###  C.1 Relation for ${\mathcal{S}}^{2}$

Let us verify that $\mathcal{S}^{2}$ maps the original theory $\mathcal{C}$
back to itself. This property of $\mathcal{S}^{2}$ is also discussed in Ref.
[Cheng and Jian, 2022] for a 2+1d bosonic topological phases $\mathcal{C}$. In
the following, we revisit the property of $\mathcal{S}^{2}$ using the gauging
approach developed in this work. The following derivation is applicable to
both bosonic and fermionic topological phase $\mathcal{C}$ We have

$\displaystyle\left({\mathcal{S}}^{2}\right){\mathcal{C}}=\frac{{\mathcal{C}}\times{\rm
U}(1)_{-nr}\times{\rm
U}(1)^{\prime}_{rn}}{\mathbb{Z}_{r}\times\mathbb{Z}_{n}}~{},$ (169)

where the $\mathbb{Z}_{r}\times\mathbb{Z}_{n}$ quotients are generated by

$\mathbb{Z}_{r}:\;v\otimes({\rm U}(1)\text{-charge }n)\otimes({\rm
U}(1)^{\prime}\text{-charge }0),\quad\mathbb{Z}_{n}:\;1\otimes({\rm
U}(1)\text{-charge }r)\otimes({\rm U}(1)^{\prime}\text{-charge }r)~{}.$ (170)

Let us denote the ${\rm U}(1)$ and ${\rm U}(1)^{\prime}$ gauge fields by $a$
and $a^{\prime}$. The Abelian Chern-Simons terms in the numerator of (178) is

$\displaystyle\frac{-nr}{4\pi}ada+\frac{nr}{4\pi}a^{\prime}da^{\prime}~{}.$
(171)

The $\mathbb{Z}_{n}$ quotient can be implemented by the following change of
variables, where we demand that $\tilde{a},\tilde{a}^{\prime}$ are properly
quantized ${\rm U}(1)$ gauge fields:

$\tilde{a}=-na,\quad\tilde{a}^{\prime}=a+a^{\prime}~{}.$ (172)

The action becomes that of $\mathbb{Z}_{r}$ gauge theory, denoted as
$(Z_{r})_{nr}$:111111 The theory $(Z_{N})_{k}$ can be described by ${\rm
U}(1)\times{\rm U}(1)$ gauge fields $a,b$ with the Chern-Simons term Maldacena
_et al._ (2001); Banks and Seiberg (2011); Kapustin and Seiberg (2014)
$\frac{k}{4\pi}ada+\frac{N}{2\pi}adb~{},$ (173) where $b$ acts as Lagrangian
multiplier that enforces $a$ to have holonomy that takes value in ${2\pi\over
N}\mathbb{Z}$. Thus $\oint a$ represents the Wilson line of the
$\mathbb{Z}_{N}$ gauge field, while $\oint b$ represents the magnetic operator
that has $e^{2\pi i/N}$ braiding with $\oint a$.

$\frac{nr}{4\pi}\tilde{a}^{\prime}d\tilde{a}^{\prime}+\frac{r}{2\pi}\tilde{a}^{\prime}d\tilde{a}~{}.$
(174)

Thus

${\mathcal{S}}^{2}{\mathcal{C}}=\frac{{\mathcal{C}}\times(Z_{r})_{nr}}{\mathbb{Z}_{r}}~{}.$
(175)

Using the dualityCordova  _et al._ (2018)

$\frac{{\mathcal{C}}\times(Z_{r})_{nr}}{\mathbb{Z}_{r}}\quad\longleftrightarrow\quad{\mathcal{C}}~{},$
(176)

we find that

${\mathcal{S}}^{2}{\mathcal{C}}={\mathcal{C}}~{}.$ (177)

###  C.2 Relation for $({\mathcal{S}\mathcal{T}})^{3}$

Let us verify that $({\mathcal{S}}{\mathcal{T}})^{3}=1$. Applying
$({\mathcal{S}}{\mathcal{T}})^{3}$ to the 2+1d topological phase
$\mathcal{C}$, we have

$\displaystyle\left({\mathcal{S}}{\mathcal{T}}\right)^{3}{\mathcal{C}}$
$\displaystyle=\frac{{\mathcal{C}}\times{\rm U}(1)_{-nr+r^{2}}\times{\rm
U}(1)^{\prime}_{-r(-n+r)+(-n+r)^{2}}\times{\rm
U}(1)^{\prime\prime}_{n(-n+r)+n^{2}}}{\mathbb{Z}_{r}\times\mathbb{Z}_{|-n+r|}\times\mathbb{Z}_{n}}$
(178) $\displaystyle=\frac{{\mathcal{C}}\times{\rm U}(1)_{(-n+r)r}\times{\rm
U}(1)^{\prime}_{-n(-n+r)}\times{\rm
U}(1)^{\prime\prime}_{nr}}{\mathbb{Z}_{r}\times\mathbb{Z}_{|-n+r|}\times\mathbb{Z}_{n}}~{},$
(179)

where the quotients are generated by

$\displaystyle\mathbb{Z}_{r}:\quad v\otimes\left({\rm U}(1)\text{-charge
}(-n+r)\right)$ (180) $\displaystyle\mathbb{Z}_{|-n+r|}:\quad\left({\rm
U}(1)\text{-charge }r\right)\otimes\left({\rm U}(1)^{\prime}\text{-charge
}n\right)$ (181) $\displaystyle\mathbb{Z}_{n}:\quad\left({\rm
U}(1)^{\prime}\text{-charge }(-n+r)\right)\otimes\left({\rm
U}(1)^{\prime\prime}\text{-charge }r\right)~{},$ (182)

where the trivial Wilson line is not written explicitly.

Let us denote the ${\rm U}(1),{\rm U}(1)^{\prime}$, and ${\rm
U}(1)^{\prime\prime}$ gauge fields by $a,a^{\prime}$, and $a^{\prime\prime}$
respectively. The Abelian Chern-Simons terms in the numerator of Eq. (178) is

$\displaystyle\frac{(-n+r)r}{4\pi}ada+\frac{-n(-n+r)}{4\pi}a^{\prime}da^{\prime}+\frac{nr}{4\pi}a^{\prime\prime}da^{\prime\prime}~{}.$
(183)

The $\mathbb{Z}_{|-n+r|}\times\mathbb{Z}_{n}$ quotient can be implemented by
the following change of variables, where we demand that
$\tilde{a},\tilde{a}^{\prime},\tilde{a}^{\prime\prime}$ are properly quantized
${\rm U}(1)$ gauge fields:

$\tilde{a}=(n-r)a,\quad\tilde{a}^{\prime}=-n(a+a^{\prime}),\quad\tilde{a}^{\prime\prime}=a+a^{\prime}+a^{\prime\prime}~{}.$
(184)

The action becomes that of ${\rm U}(1)_{1}\times(Z_{r})_{nr}$:

$\frac{1}{4\pi}(\tilde{a}+\tilde{a}^{\prime})d(\tilde{a}+\tilde{a}^{\prime})+\frac{nr}{4\pi}\tilde{a}^{\prime\prime}d\tilde{a}^{\prime\prime}+\frac{r}{2\pi}\tilde{a}^{\prime\prime}d\tilde{a}^{\prime}~{}.$
(185)

Thus

$\left({\mathcal{S}}{\mathcal{T}}\right)^{3}{\mathcal{C}}={\rm
U}(1)_{1}\times\frac{{\mathcal{C}}\times(Z_{r})_{nr}}{\mathbb{Z}_{r}}~{}.$
(186)

Using the dualityCordova  _et al._ (2018)

$\frac{{\mathcal{C}}\times(Z_{r})_{nr}}{\mathbb{Z}_{r}}\quad\longleftrightarrow\quad{\mathcal{C}}~{},$
(187)

we find that

$\left({\mathcal{S}}{\mathcal{T}}\right)^{3}{\mathcal{C}}={\rm
U}(1)_{1}\times{\mathcal{C}}~{}.$ (188)

Thus we find that $\left({\mathcal{S}}{\mathcal{T}}\right)^{3}=1$ up to the
invertible Chern-Simons theory ${\rm U}(1)_{1}$ (which is equivalent to the
gravitational Chern-Simons term with chiral central charge 1), in agreement
with the result in Ref. [Witten, 2003].

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